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Theorem nbssntrs 26250
Description: IF  X and  Y are not on the same side, and  Y and  Z are not on the same side then 
X and  Z are on the same side. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
Hypotheses
Ref Expression
bsstrs.1  |-  P  =  (PPoints `  G )
bsstrs.2  |-  L  =  (PLines `  G )
bsstrs.3  |-  S  =  ( seg `  G
)
bsstrs.4  |-  ( ph  ->  G  e. Ibg )
bsstrs.5  |-  ( ph  ->  M  e.  L )
nbssntrs.6  |-  ( ph  ->  X  e.  ( P 
\  M ) )
nbssntrs.7  |-  ( ph  ->  Y  e.  ( P 
\  M ) )
nbssntrs.8  |-  ( ph  ->  Z  e.  ( P 
\  M ) )
nbssntrs.9  |-  ( ph  ->  ( ( X S Y )  i^i  M
)  =/=  (/) )
nbssntrs.10  |-  ( ph  ->  ( ( Y S Z )  i^i  M
)  =/=  (/) )
Assertion
Ref Expression
nbssntrs  |-  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) )

Proof of Theorem nbssntrs
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bsstrs.1 . . . . . 6  |-  P  =  (PPoints `  G )
2 bsstrs.3 . . . . . 6  |-  S  =  ( seg `  G
)
3 bsstrs.4 . . . . . . 7  |-  ( ph  ->  G  e. Ibg )
43adantl 452 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  G  e. Ibg )
5 nbssntrs.6 . . . . . . . 8  |-  ( ph  ->  X  e.  ( P 
\  M ) )
6 eldifi 3311 . . . . . . . 8  |-  ( X  e.  ( P  \  M )  ->  X  e.  P )
75, 6syl 15 . . . . . . 7  |-  ( ph  ->  X  e.  P )
87adantl 452 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  X  e.  P
)
9 eqid 2296 . . . . . 6  |-  (btw `  G )  =  (btw
`  G )
10 nbssntrs.8 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( P 
\  M ) )
11 eldifi 3311 . . . . . . . 8  |-  ( Z  e.  ( P  \  M )  ->  Z  e.  P )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  Z  e.  P )
1312adantl 452 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  Z  e.  P
)
14 df-ne 2461 . . . . . . . . 9  |-  ( X  =/=  Z  <->  -.  X  =  Z )
1514biimpri 197 . . . . . . . 8  |-  ( -.  X  =  Z  ->  X  =/=  Z )
16153ad2ant2 977 . . . . . . 7  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  X  =/=  Z )
1716adantr 451 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  X  =/=  Z
)
181, 2, 4, 8, 9, 13, 17sgplpte21 26235 . . . . 5  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( X S Z )  =  {
x  e.  P  | 
( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z ) } )
1918ineq1d 3382 . . . 4  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( X S Z )  i^i 
M )  =  ( { x  e.  P  |  ( x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) }  i^i  M
) )
20 inrab2 3454 . . . . 5  |-  ( { x  e.  P  | 
( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z ) }  i^i  M )  =  { x  e.  ( P  i^i  M )  |  ( x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) }
21 incom 3374 . . . . . . . . . . 11  |-  ( M  i^i  ( X (btw
`  G ) Z ) )  =  ( ( X (btw `  G ) Z )  i^i  M )
22 bsstrs.2 . . . . . . . . . . . 12  |-  L  =  (PLines `  G )
23 bsstrs.5 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  L )
2423adantl 452 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  M  e.  L
)
255adantl 452 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  X  e.  ( P  \  M ) )
26 nbssntrs.7 . . . . . . . . . . . . 13  |-  ( ph  ->  Y  e.  ( P 
\  M ) )
2726adantl 452 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  Y  e.  ( P  \  M ) )
2810adantl 452 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  Z  e.  ( P  \  M ) )
29 nbssntrs.9 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X S Y )  i^i  M
)  =/=  (/) )
303adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  G  e. Ibg )
317adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  X  e.  P )
32 eldifi 3311 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( Y  e.  ( P  \  M )  ->  Y  e.  P )
3326, 32syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  Y  e.  P )
3433adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Y  e.  P )
35 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( X  =/=  Y  <->  -.  X  =  Y )
3635biimpri 197 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  X  =  Y  ->  X  =/=  Y )
37363ad2ant3 978 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  X  =/=  Y )
3837adantl 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  X  =/=  Y )
391, 2, 30, 31, 9, 34, 38sgplpte21 26235 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y ) } )
4039ineq1d 3382 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( X S Y )  i^i  M )  =  ( { z  e.  P  |  ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y ) }  i^i  M ) )
41 inrab2 3454 . . . . . . . . . . . . . . . . . 18  |-  ( { z  e.  P  | 
( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y ) }  i^i  M )  =  { z  e.  ( P  i^i  M )  |  ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y ) }
4240, 41syl6eq 2344 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( X S Y )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( X (btw
`  G ) Y )  \/  z  =  X  \/  z  =  Y ) } )
43 neeq1 2467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X S Y )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( X (btw
`  G ) Y )  \/  z  =  X  \/  z  =  Y ) }  ->  ( ( ( X S Y )  i^i  M
)  =/=  (/)  <->  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( X (btw
`  G ) Y )  \/  z  =  X  \/  z  =  Y ) }  =/=  (/) ) )
44 rabn0 3487 . . . . . . . . . . . . . . . . . . 19  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y ) }  =/=  (/)  <->  E. z  e.  ( P  i^i  M ) ( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y )
)
45 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z  e.  ( P  i^i  M )  <->  ( z  e.  P  /\  z  e.  M ) )
46 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  ( ( X (btw `  G ) Y )  i^i  M
)  <->  ( z  e.  ( X (btw `  G ) Y )  /\  z  e.  M
) )
47 ne0i 3474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  ( ( X (btw `  G ) Y )  i^i  M
)  ->  ( ( X (btw `  G ) Y )  i^i  M
)  =/=  (/) )
4847a1d 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  ( ( X (btw `  G ) Y )  i^i  M
)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
4946, 48sylbir 204 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( z  e.  ( X (btw `  G ) Y )  /\  z  e.  M )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
5049expcom 424 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z  e.  M  ->  (
z  e.  ( X (btw `  G ) Y )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
5150adantl 452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( z  e.  P  /\  z  e.  M )  ->  ( z  e.  ( X (btw `  G
) Y )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
5245, 51sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  e.  ( P  i^i  M )  ->  ( z  e.  ( X (btw `  G ) Y )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
5352com12 27 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  e.  ( X (btw
`  G ) Y )  ->  ( z  e.  ( P  i^i  M
)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
54 eleq1 2356 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  =  X  ->  (
z  e.  ( P  i^i  M )  <->  X  e.  ( P  i^i  M ) ) )
55 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( X  e.  ( P  i^i  M )  <->  ( X  e.  P  /\  X  e.  M ) )
56 eldifn 3312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( X  e.  ( P  \  M )  ->  -.  X  e.  M )
575, 56syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  -.  X  e.  M
)
5857pm2.21d 98 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( X  e.  M  ->  ( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
5958adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( X  e.  M  ->  ( ( X (btw `  G ) Y )  i^i  M )  =/=  (/) ) )
6059com12 27 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( X  e.  M  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
6160adantl 452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( X  e.  P  /\  X  e.  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
6255, 61sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( X  e.  ( P  i^i  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
6354, 62syl6bi 219 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  =  X  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
64 eleq1 2356 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  =  Y  ->  (
z  e.  ( P  i^i  M )  <->  Y  e.  ( P  i^i  M ) ) )
65 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Y  e.  ( P  i^i  M )  <->  ( Y  e.  P  /\  Y  e.  M ) )
66 eldifn 3312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( Y  e.  ( P  \  M )  ->  -.  Y  e.  M )
6726, 66syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  -.  Y  e.  M
)
6867pm2.21d 98 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( Y  e.  M  ->  ( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
6968adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( Y  e.  M  ->  ( ( X (btw `  G ) Y )  i^i  M )  =/=  (/) ) )
7069com12 27 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Y  e.  M  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
7170adantl 452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Y  e.  P  /\  Y  e.  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
7265, 71sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Y  e.  ( P  i^i  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
7364, 72syl6bi 219 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  =  Y  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
7453, 63, 733jaoi 1245 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y )  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
7574com12 27 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( P  i^i  M )  ->  ( (
z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
7675rexlimiv 2674 . . . . . . . . . . . . . . . . . . 19  |-  ( E. z  e.  ( P  i^i  M ) ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
7744, 76sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y ) }  =/=  (/)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) )
7843, 77syl6bi 219 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X S Y )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( X (btw
`  G ) Y )  \/  z  =  X  \/  z  =  Y ) }  ->  ( ( ( X S Y )  i^i  M
)  =/=  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
7942, 78syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( X S Y )  i^i  M
)  =/=  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =/=  (/) ) ) )
8079pm2.43a 45 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( X S Y )  i^i  M
)  =/=  (/)  ->  (
( X (btw `  G ) Y )  i^i  M )  =/=  (/) ) )
8180ex 423 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( ( X S Y )  i^i  M
)  =/=  (/)  ->  (
( X (btw `  G ) Y )  i^i  M )  =/=  (/) ) ) )
8229, 81mpid 37 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( X (btw `  G ) Y )  i^i  M )  =/=  (/) ) )
8382impcom 419 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( X (btw `  G ) Y )  i^i  M
)  =/=  (/) )
84 nbssntrs.10 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( Y S Z )  i^i  M
)  =/=  (/) )
8512adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Z  e.  P )
86 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Y  =/=  Z  <->  -.  Y  =  Z )
8786biimpri 197 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  Y  =  Z  ->  Y  =/=  Z )
88873ad2ant1 976 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  Y  =/=  Z )
8988adantl 452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Y  =/=  Z )
901, 2, 30, 34, 9, 85, 89sgplpte21 26235 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( Y S Z )  =  { z  e.  P  |  ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z ) } )
9190ineq1d 3382 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( Y S Z )  i^i  M )  =  ( { z  e.  P  |  ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z ) }  i^i  M ) )
92 inrab2 3454 . . . . . . . . . . . . . . . . . . 19  |-  ( { z  e.  P  | 
( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z ) }  i^i  M )  =  { z  e.  ( P  i^i  M )  |  ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z ) }
9391, 92syl6eq 2344 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( Y S Z )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( Y (btw
`  G ) Z )  \/  z  =  Y  \/  z  =  Z ) } )
94 neeq1 2467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( Y S Z )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( Y (btw
`  G ) Z )  \/  z  =  Y  \/  z  =  Z ) }  ->  ( ( ( Y S Z )  i^i  M
)  =/=  (/)  <->  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( Y (btw
`  G ) Z )  \/  z  =  Y  \/  z  =  Z ) }  =/=  (/) ) )
95 rabn0 3487 . . . . . . . . . . . . . . . . . . . 20  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z ) }  =/=  (/)  <->  E. z  e.  ( P  i^i  M ) ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z )
)
96 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  ( ( Y (btw `  G ) Z )  i^i  M
)  <->  ( z  e.  ( Y (btw `  G ) Z )  /\  z  e.  M
) )
97 ne0i 3474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( z  e.  ( ( Y (btw `  G ) Z )  i^i  M
)  ->  ( ( Y (btw `  G ) Z )  i^i  M
)  =/=  (/) )
9897a1ii 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
z  e.  ( ( Y (btw `  G
) Z )  i^i 
M )  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
9998com3r 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  ( ( Y (btw `  G ) Z )  i^i  M
)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
10096, 99sylbir 204 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( z  e.  ( Y (btw `  G ) Z )  /\  z  e.  M )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
101100expcom 424 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z  e.  M  ->  (
z  e.  ( Y (btw `  G ) Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
102101adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( z  e.  P  /\  z  e.  M )  ->  ( z  e.  ( Y (btw `  G
) Z )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
10345, 102sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z  e.  ( P  i^i  M )  ->  ( z  e.  ( Y (btw `  G ) Z )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
104103com12 27 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  e.  ( Y (btw
`  G ) Z )  ->  ( z  e.  ( P  i^i  M
)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
10567pm2.21d 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( Y  e.  M  ->  ( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
106105adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( Y  e.  M  ->  ( ( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
107106com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Y  e.  M  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
108107adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( Y  e.  P  /\  Y  e.  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
10965, 108sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Y  e.  ( P  i^i  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
11064, 109syl6bi 219 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  =  Y  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
111 eleq1 2356 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z  =  Z  ->  (
z  e.  ( P  i^i  M )  <->  Z  e.  ( P  i^i  M ) ) )
112 elin 3371 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Z  e.  ( P  i^i  M )  <->  ( Z  e.  P  /\  Z  e.  M ) )
113 eldifn 3312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( Z  e.  ( P  \  M )  ->  -.  Z  e.  M )
11410, 113syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  -.  Z  e.  M
)
115114pm2.21d 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( Z  e.  M  ->  ( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
116115adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( Z  e.  M  ->  ( ( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
117116com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Z  e.  M  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
118117adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( Z  e.  P  /\  Z  e.  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
119112, 118sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Z  e.  ( P  i^i  M )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
120111, 119syl6bi 219 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  =  Z  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
121104, 110, 1203jaoi 1245 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z )  ->  (
z  e.  ( P  i^i  M )  -> 
( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
122121com12 27 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  e.  ( P  i^i  M )  ->  ( (
z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
123122rexlimiv 2674 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. z  e.  ( P  i^i  M ) ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
12495, 123sylbi 187 . . . . . . . . . . . . . . . . . . 19  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z ) }  =/=  (/)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
12594, 124syl6bi 219 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( Y S Z )  i^i  M )  =  { z  e.  ( P  i^i  M
)  |  ( z  e.  ( Y (btw
`  G ) Z )  \/  z  =  Y  \/  z  =  Z ) }  ->  ( ( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
12693, 125syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( ( Y S Z )  i^i 
M )  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
127126pm2.43a 45 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
128127ex 423 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) ) )
12984, 128mpid 37 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) ) )
13084, 129mpid 37 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( Y (btw `  G ) Z )  i^i  M )  =/=  (/) ) )
131130impcom 419 . . . . . . . . . . . 12  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( Y (btw `  G ) Z )  i^i  M
)  =/=  (/) )
1321, 22, 9, 4, 24, 25, 27, 28, 83, 131nbssntr 26232 . . . . . . . . . . 11  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( X (btw `  G ) Z )  i^i  M
)  =  (/) )
13321, 132syl5eq 2340 . . . . . . . . . 10  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( M  i^i  ( X (btw `  G
) Z ) )  =  (/) )
134 disj 3508 . . . . . . . . . 10  |-  ( ( M  i^i  ( X (btw `  G ) Z ) )  =  (/) 
<-> 
A. x  e.  M  -.  x  e.  ( X (btw `  G ) Z ) )
135133, 134sylib 188 . . . . . . . . 9  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  M  -.  x  e.  ( X (btw `  G ) Z ) )
136 risset 2603 . . . . . . . . . . . 12  |-  ( X  e.  M  <->  E. x  e.  M  x  =  X )
13757, 136sylnib 295 . . . . . . . . . . 11  |-  ( ph  ->  -.  E. x  e.  M  x  =  X )
138137adantl 452 . . . . . . . . . 10  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  -.  E. x  e.  M  x  =  X )
139 ralnex 2566 . . . . . . . . . 10  |-  ( A. x  e.  M  -.  x  =  X  <->  -.  E. x  e.  M  x  =  X )
140138, 139sylibr 203 . . . . . . . . 9  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  M  -.  x  =  X
)
141114adantl 452 . . . . . . . . . . 11  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  -.  Z  e.  M )
142 risset 2603 . . . . . . . . . . 11  |-  ( Z  e.  M  <->  E. x  e.  M  x  =  Z )
143141, 142sylnib 295 . . . . . . . . . 10  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  -.  E. x  e.  M  x  =  Z )
144 ralnex 2566 . . . . . . . . . 10  |-  ( A. x  e.  M  -.  x  =  Z  <->  -.  E. x  e.  M  x  =  Z )
145143, 144sylibr 203 . . . . . . . . 9  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  M  -.  x  =  Z
)
146 r19.26-3 2690 . . . . . . . . 9  |-  ( A. x  e.  M  ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
)  <->  ( A. x  e.  M  -.  x  e.  ( X (btw `  G ) Z )  /\  A. x  e.  M  -.  x  =  X  /\  A. x  e.  M  -.  x  =  Z ) )
147135, 140, 145, 146syl3anbrc 1136 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  M  ( -.  x  e.  ( X (btw `  G
) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) )
148 inss2 3403 . . . . . . . . . . . . 13  |-  ( P  i^i  M )  C_  M
149148a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  i^i  M
)  C_  M )
150149sseld 3192 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( P  i^i  M )  ->  x  e.  M
) )
151150imim1d 69 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  M  ->  ( -.  x  e.  ( X
(btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
) )  ->  (
x  e.  ( P  i^i  M )  -> 
( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) ) ) )
152151adantl 452 . . . . . . . . 9  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( x  e.  M  ->  ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
) )  ->  (
x  e.  ( P  i^i  M )  -> 
( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) ) ) )
153152ralimdv2 2636 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( A. x  e.  M  ( -.  x  e.  ( X
(btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
)  ->  A. x  e.  ( P  i^i  M
) ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) ) )
154147, 153mpd 14 . . . . . . 7  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  ( P  i^i  M ) ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) )
155 3ioran 950 . . . . . . . . 9  |-  ( -.  ( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z )  <->  ( -.  x  e.  ( X (btw `  G
) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) )
156155a1i 10 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( -.  (
x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z )  <->  ( -.  x  e.  ( X
(btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
) ) )
157156ralbidv 2576 . . . . . . 7  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( A. x  e.  ( P  i^i  M
)  -.  ( x  e.  ( X (btw
`  G ) Z )  \/  x  =  X  \/  x  =  Z )  <->  A. x  e.  ( P  i^i  M
) ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) ) )
158154, 157mpbird 223 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  ( P  i^i  M )  -.  ( x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) )
159 rabeq0 3489 . . . . . 6  |-  ( { x  e.  ( P  i^i  M )  |  ( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z ) }  =  (/)  <->  A. x  e.  ( P  i^i  M
)  -.  ( x  e.  ( X (btw
`  G ) Z )  \/  x  =  X  \/  x  =  Z ) )
160158, 159sylibr 203 . . . . 5  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  { x  e.  ( P  i^i  M
)  |  ( x  e.  ( X (btw
`  G ) Z )  \/  x  =  X  \/  x  =  Z ) }  =  (/) )
16120, 160syl5eq 2340 . . . 4  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( { x  e.  P  |  (
x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) }  i^i  M )  =  (/) )
16219, 161eqtrd 2328 . . 3  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( X S Z )  i^i 
M )  =  (/) )
1631623exp1 1167 . 2  |-  ( -.  Y  =  Z  -> 
( -.  X  =  Z  ->  ( -.  X  =  Y  ->  (
ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) ) )
164 oveq1 5881 . . . . . . . 8  |-  ( Y  =  Z  ->  ( Y S Z )  =  ( Z S Z ) )
165164ineq1d 3382 . . . . . . 7  |-  ( Y  =  Z  ->  (
( Y S Z )  i^i  M )  =  ( ( Z S Z )  i^i 
M ) )
166165neeq1d 2472 . . . . . 6  |-  ( Y  =  Z  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  <->  ( ( Z S Z )  i^i 
M )  =/=  (/) ) )
1671, 2, 3, 12sgplpte22 26241 . . . . . . . . . . 11  |-  ( ph  ->  ( Z S Z )  =  { Z } )
168167ineq1d 3382 . . . . . . . . . 10  |-  ( ph  ->  ( ( Z S Z )  i^i  M
)  =  ( { Z }  i^i  M
) )
169168neeq1d 2472 . . . . . . . . 9  |-  ( ph  ->  ( ( ( Z S Z )  i^i 
M )  =/=  (/)  <->  ( { Z }  i^i  M )  =/=  (/) ) )
170 df-ne 2461 . . . . . . . . . 10  |-  ( ( { Z }  i^i  M )  =/=  (/)  <->  -.  ( { Z }  i^i  M
)  =  (/) )
171 incom 3374 . . . . . . . . . . . 12  |-  ( { Z }  i^i  M
)  =  ( M  i^i  { Z }
)
172171eqeq1i 2303 . . . . . . . . . . 11  |-  ( ( { Z }  i^i  M )  =  (/)  <->  ( M  i^i  { Z } )  =  (/) )
173 disjsn 3706 . . . . . . . . . . . 12  |-  ( ( M  i^i  { Z } )  =  (/)  <->  -.  Z  e.  M )
174 notnot2 104 . . . . . . . . . . . 12  |-  ( -. 
-.  Z  e.  M  ->  Z  e.  M )
175173, 174sylnbi 297 . . . . . . . . . . 11  |-  ( -.  ( M  i^i  { Z } )  =  (/)  ->  Z  e.  M )
176172, 175sylnbi 297 . . . . . . . . . 10  |-  ( -.  ( { Z }  i^i  M )  =  (/)  ->  Z  e.  M )
177170, 176sylbi 187 . . . . . . . . 9  |-  ( ( { Z }  i^i  M )  =/=  (/)  ->  Z  e.  M )
178169, 177syl6bi 219 . . . . . . . 8  |-  ( ph  ->  ( ( ( Z S Z )  i^i 
M )  =/=  (/)  ->  Z  e.  M ) )
179114pm2.21d 98 . . . . . . . 8  |-  ( ph  ->  ( Z  e.  M  ->  ( ( X S Z )  i^i  M
)  =  (/) ) )
180178, 179syld 40 . . . . . . 7  |-  ( ph  ->  ( ( ( Z S Z )  i^i 
M )  =/=  (/)  ->  (
( X S Z )  i^i  M )  =  (/) ) )
181180com12 27 . . . . . 6  |-  ( ( ( Z S Z )  i^i  M )  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) ) )
182166, 181syl6bi 219 . . . . 5  |-  ( Y  =  Z  ->  (
( ( Y S Z )  i^i  M
)  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) )
183182com3l 75 . . . 4  |-  ( ( ( Y S Z )  i^i  M )  =/=  (/)  ->  ( ph  ->  ( Y  =  Z  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) )
18484, 183mpcom 32 . . 3  |-  ( ph  ->  ( Y  =  Z  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
185184com12 27 . 2  |-  ( Y  =  Z  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
1861, 2, 3, 7sgplpte22 26241 . . . 4  |-  ( ph  ->  ( X S X )  =  { X } )
187 incom 3374 . . . . . 6  |-  ( { X }  i^i  M
)  =  ( M  i^i  { X }
)
188 disjsn 3706 . . . . . . 7  |-  ( ( M  i^i  { X } )  =  (/)  <->  -.  X  e.  M )
18957, 188sylibr 203 . . . . . 6  |-  ( ph  ->  ( M  i^i  { X } )  =  (/) )
190187, 189syl5eq 2340 . . . . 5  |-  ( ph  ->  ( { X }  i^i  M )  =  (/) )
191 ineq1 3376 . . . . . 6  |-  ( ( X S X )  =  { X }  ->  ( ( X S X )  i^i  M
)  =  ( { X }  i^i  M
) )
192191eqeq1d 2304 . . . . 5  |-  ( ( X S X )  =  { X }  ->  ( ( ( X S X )  i^i 
M )  =  (/)  <->  ( { X }  i^i  M
)  =  (/) ) )
193190, 192syl5ibr 212 . . . 4  |-  ( ( X S X )  =  { X }  ->  ( ph  ->  (
( X S X )  i^i  M )  =  (/) ) )
194186, 193mpcom 32 . . 3  |-  ( ph  ->  ( ( X S X )  i^i  M
)  =  (/) )
195 oveq2 5882 . . . . . 6  |-  ( Z  =  X  ->  ( X S Z )  =  ( X S X ) )
196195eqcoms 2299 . . . . 5  |-  ( X  =  Z  ->  ( X S Z )  =  ( X S X ) )
197196ineq1d 3382 . . . 4  |-  ( X  =  Z  ->  (
( X S Z )  i^i  M )  =  ( ( X S X )  i^i 
M ) )
198197eqeq1d 2304 . . 3  |-  ( X  =  Z  ->  (
( ( X S Z )  i^i  M
)  =  (/)  <->  ( ( X S X )  i^i 
M )  =  (/) ) )
199194, 198syl5ibr 212 . 2  |-  ( X  =  Z  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
200 oveq1 5881 . . . . . . . 8  |-  ( X  =  Y  ->  ( X S Y )  =  ( Y S Y ) )
201200ineq1d 3382 . . . . . . 7  |-  ( X  =  Y  ->  (
( X S Y )  i^i  M )  =  ( ( Y S Y )  i^i 
M ) )
202201neeq1d 2472 . . . . . 6  |-  ( X  =  Y  ->  (
( ( X S Y )  i^i  M
)  =/=  (/)  <->  ( ( Y S Y )  i^i 
M )  =/=  (/) ) )
2031, 2, 3, 33sgplpte22 26241 . . . . . . . 8  |-  ( ph  ->  ( Y S Y )  =  { Y } )
204 ineq1 3376 . . . . . . . . . . 11  |-  ( ( Y S Y )  =  { Y }  ->  ( ( Y S Y )  i^i  M
)  =  ( { Y }  i^i  M
) )
205204neeq1d 2472 . . . . . . . . . 10  |-  ( ( Y S Y )  =  { Y }  ->  ( ( ( Y S Y )  i^i 
M )  =/=  (/)  <->  ( { Y }  i^i  M )  =/=  (/) ) )
206 incom 3374 . . . . . . . . . . . 12  |-  ( { Y }  i^i  M
)  =  ( M  i^i  { Y }
)
207206neeq1i 2469 . . . . . . . . . . 11  |-  ( ( { Y }  i^i  M )  =/=  (/)  <->  ( M  i^i  { Y } )  =/=  (/) )
208 disjsn 3706 . . . . . . . . . . . . . 14  |-  ( ( M  i^i  { Y } )  =  (/)  <->  -.  Y  e.  M )
209208bicomi 193 . . . . . . . . . . . . 13  |-  ( -.  Y  e.  M  <->  ( M  i^i  { Y } )  =  (/) )
210209necon3bbii 2490 . . . . . . . . . . . 12  |-  ( -. 
-.  Y  e.  M  <->  ( M  i^i  { Y } )  =/=  (/) )
211 notnot2 104 . . . . . . . . . . . . 13  |-  ( -. 
-.  Y  e.  M  ->  Y  e.  M )
21267pm2.21d 98 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Y  e.  M  ->  ( ( X S Z )  i^i  M
)  =  (/) ) )
213212com12 27 . . . . . . . . . . . . 13  |-  ( Y  e.  M  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
214211, 213syl 15 . . . . . . . . . . . 12  |-  ( -. 
-.  Y  e.  M  ->  ( ph  ->  (
( X S Z )  i^i  M )  =  (/) ) )
215210, 214sylbir 204 . . . . . . . . . . 11  |-  ( ( M  i^i  { Y } )  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
216207, 215sylbi 187 . . . . . . . . . 10  |-  ( ( { Y }  i^i  M )  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
217205, 216syl6bi 219 . . . . . . . . 9  |-  ( ( Y S Y )  =  { Y }  ->  ( ( ( Y S Y )  i^i 
M )  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) )
218217com23 72 . . . . . . . 8  |-  ( ( Y S Y )  =  { Y }  ->  ( ph  ->  (
( ( Y S Y )  i^i  M
)  =/=  (/)  ->  (
( X S Z )  i^i  M )  =  (/) ) ) )
219203, 218mpcom 32 . . . . . . 7  |-  ( ph  ->  ( ( ( Y S Y )  i^i 
M )  =/=  (/)  ->  (
( X S Z )  i^i  M )  =  (/) ) )
220219com12 27 . . . . . 6  |-  ( ( ( Y S Y )  i^i  M )  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) ) )
221202, 220syl6bi 219 . . . . 5  |-  ( X  =  Y  ->  (
( ( X S Y )  i^i  M
)  =/=  (/)  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) )
222221com3l 75 . . . 4  |-  ( ( ( X S Y )  i^i  M )  =/=  (/)  ->  ( ph  ->  ( X  =  Y  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) )
22329, 222mpcom 32 . . 3  |-  ( ph  ->  ( X  =  Y  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
224223com12 27 . 2  |-  ( X  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
225163, 185, 199, 224pm2.61iii 159 1  |-  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  btwcbtw 26209  Ibgcibg 26210   segcseg 26233
This theorem is referenced by:  pdiveql  26271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ibg2 26212  df-seg2 26234
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