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Theorem bsstrs 26249
Description: Being on the same side is a transitive relation. Segment version of bsstr 26231. (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 )
bsstrs.6  |-  ( ph  ->  X  e.  ( P 
\  M ) )
bsstrs.7  |-  ( ph  ->  Y  e.  ( P 
\  M ) )
bsstrs.8  |-  ( ph  ->  Z  e.  ( P 
\  M ) )
bsstrs.9  |-  ( ph  ->  ( ( X S Y )  i^i  M
)  =  (/) )
bsstrs.10  |-  ( ph  ->  ( ( Y S Z )  i^i  M
)  =  (/) )
Assertion
Ref Expression
bsstrs  |-  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) )

Proof of Theorem bsstrs
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bsstrs.10 . . 3  |-  ( ph  ->  ( ( Y S Z )  i^i  M
)  =  (/) )
2 oveq1 5881 . . . . 5  |-  ( X  =  Y  ->  ( X S Z )  =  ( Y S Z ) )
32ineq1d 3382 . . . 4  |-  ( X  =  Y  ->  (
( X S Z )  i^i  M )  =  ( ( Y S Z )  i^i 
M ) )
43eqeq1d 2304 . . 3  |-  ( X  =  Y  ->  (
( ( X S Z )  i^i  M
)  =  (/)  <->  ( ( Y S Z )  i^i 
M )  =  (/) ) )
51, 4syl5ibr 212 . 2  |-  ( X  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
6 bsstrs.1 . . . . . . . . 9  |-  P  =  (PPoints `  G )
7 bsstrs.3 . . . . . . . . 9  |-  S  =  ( seg `  G
)
8 bsstrs.4 . . . . . . . . 9  |-  ( ph  ->  G  e. Ibg )
9 bsstrs.6 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( P 
\  M ) )
10 eldifi 3311 . . . . . . . . . 10  |-  ( X  e.  ( P  \  M )  ->  X  e.  P )
119, 10syl 15 . . . . . . . . 9  |-  ( ph  ->  X  e.  P )
126, 7, 8, 11sgplpte22 26241 . . . . . . . 8  |-  ( ph  ->  ( X S X )  =  { X } )
1312ineq1d 3382 . . . . . . 7  |-  ( ph  ->  ( ( X S X )  i^i  M
)  =  ( { X }  i^i  M
) )
14 incom 3374 . . . . . . . 8  |-  ( { X }  i^i  M
)  =  ( M  i^i  { X }
)
15 eldifn 3312 . . . . . . . . . 10  |-  ( X  e.  ( P  \  M )  ->  -.  X  e.  M )
169, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  -.  X  e.  M
)
17 disjsn 3706 . . . . . . . . 9  |-  ( ( M  i^i  { X } )  =  (/)  <->  -.  X  e.  M )
1816, 17sylibr 203 . . . . . . . 8  |-  ( ph  ->  ( M  i^i  { X } )  =  (/) )
1914, 18syl5eq 2340 . . . . . . 7  |-  ( ph  ->  ( { X }  i^i  M )  =  (/) )
2013, 19eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( X S X )  i^i  M
)  =  (/) )
2120a1i 10 . . . . 5  |-  ( -.  X  =  Y  -> 
( ph  ->  ( ( X S X )  i^i  M )  =  (/) ) )
22 oveq2 5882 . . . . . . . 8  |-  ( Z  =  X  ->  ( X S Z )  =  ( X S X ) )
2322ineq1d 3382 . . . . . . 7  |-  ( Z  =  X  ->  (
( X S Z )  i^i  M )  =  ( ( X S X )  i^i 
M ) )
2423eqeq1d 2304 . . . . . 6  |-  ( Z  =  X  ->  (
( ( X S Z )  i^i  M
)  =  (/)  <->  ( ( X S X )  i^i 
M )  =  (/) ) )
2524imbi2d 307 . . . . 5  |-  ( Z  =  X  ->  (
( ph  ->  ( ( X S Z )  i^i  M )  =  (/) )  <->  ( ph  ->  ( ( X S X )  i^i  M )  =  (/) ) ) )
2621, 25syl5ibr 212 . . . 4  |-  ( Z  =  X  ->  ( -.  X  =  Y  ->  ( ph  ->  (
( X S Z )  i^i  M )  =  (/) ) ) )
2726eqcoms 2299 . . 3  |-  ( X  =  Z  ->  ( -.  X  =  Y  ->  ( ph  ->  (
( X S Z )  i^i  M )  =  (/) ) ) )
28 bsstrs.9 . . . . . . . 8  |-  ( ph  ->  ( ( X S Y )  i^i  M
)  =  (/) )
29 oveq2 5882 . . . . . . . . . 10  |-  ( Z  =  Y  ->  ( X S Z )  =  ( X S Y ) )
3029ineq1d 3382 . . . . . . . . 9  |-  ( Z  =  Y  ->  (
( X S Z )  i^i  M )  =  ( ( X S Y )  i^i 
M ) )
3130eqeq1d 2304 . . . . . . . 8  |-  ( Z  =  Y  ->  (
( ( X S Z )  i^i  M
)  =  (/)  <->  ( ( X S Y )  i^i 
M )  =  (/) ) )
3228, 31syl5ibr 212 . . . . . . 7  |-  ( Z  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) )
3332a1d 22 . . . . . 6  |-  ( Z  =  Y  ->  ( -.  X  =  Y  ->  ( ph  ->  (
( X S Z )  i^i  M )  =  (/) ) ) )
3433a1d 22 . . . . 5  |-  ( Z  =  Y  ->  ( -.  X  =  Z  ->  ( -.  X  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) ) ) ) )
3534eqcoms 2299 . . . 4  |-  ( Y  =  Z  ->  ( -.  X  =  Z  ->  ( -.  X  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) ) ) ) )
368adantl 452 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  G  e. Ibg )
3711adantl 452 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  X  e.  P
)
38 eqid 2296 . . . . . . . 8  |-  (btw `  G )  =  (btw
`  G )
39 bsstrs.8 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( P 
\  M ) )
40 eldifi 3311 . . . . . . . . . 10  |-  ( Z  e.  ( P  \  M )  ->  Z  e.  P )
4139, 40syl 15 . . . . . . . . 9  |-  ( ph  ->  Z  e.  P )
4241adantl 452 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  Z  e.  P
)
43 df-ne 2461 . . . . . . . . . . 11  |-  ( X  =/=  Z  <->  -.  X  =  Z )
4443biimpri 197 . . . . . . . . . 10  |-  ( -.  X  =  Z  ->  X  =/=  Z )
45443ad2ant2 977 . . . . . . . . 9  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  X  =/=  Z )
4645adantr 451 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  X  =/=  Z
)
476, 7, 36, 37, 38, 42, 46sgplpte21 26235 . . . . . . 7  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( X S Z )  =  {
x  e.  P  | 
( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z ) } )
4847ineq1d 3382 . . . . . 6  |-  ( ( ( -.  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
) )
49 inrab2 3454 . . . . . . 7  |-  ( { 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 ) }
50 incom 3374 . . . . . . . . . . . . . 14  |-  ( M  i^i  ( X (btw
`  G ) Z ) )  =  ( ( X (btw `  G ) Z )  i^i  M )
51 bsstrs.2 . . . . . . . . . . . . . . 15  |-  L  =  (PLines `  G )
528adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  G  e. Ibg )
53 bsstrs.5 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  L )
5453adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  M  e.  L )
559adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  X  e.  ( P  \  M
) )
56 bsstrs.7 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Y  e.  ( P 
\  M ) )
5756adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Y  e.  ( P  \  M
) )
5839adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Z  e.  ( P  \  M
) )
5911adantr 451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  X  e.  P )
60 eldifi 3311 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Y  e.  ( P  \  M )  ->  Y  e.  P )
6156, 60syl 15 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  Y  e.  P )
6261adantr 451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Y  e.  P )
63 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( X  =/=  Y  <->  -.  X  =  Y )
6463biimpri 197 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  X  =  Y  ->  X  =/=  Y )
65643ad2ant3 978 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  X  =/=  Y )
6665adantl 452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  X  =/=  Y )
676, 7, 52, 59, 38, 62, 66sgplpte21 26235 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y ) } )
6867ineq1d 3382 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
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 ) )
69 inrab2 3454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { 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 ) }
7068, 69syl6eq 2344 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
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 ) } )
71 eqtr2 2314 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( 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 ) }  =  (/) )
72 rabeq0 3489 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y ) }  =  (/)  <->  A. z  e.  ( P  i^i  M
)  -.  ( z  e.  ( X (btw
`  G ) Y )  \/  z  =  X  \/  z  =  Y ) )
73 3ioran 950 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  ( z  e.  ( X (btw `  G
) Y )  \/  z  =  X  \/  z  =  Y )  <->  ( -.  z  e.  ( X (btw `  G
) Y )  /\  -.  z  =  X  /\  -.  z  =  Y ) )
7473ralbii 2580 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A. z  e.  ( P  i^i  M )  -.  (
z  e.  ( X (btw `  G ) Y )  \/  z  =  X  \/  z  =  Y )  <->  A. z  e.  ( P  i^i  M
) ( -.  z  e.  ( X (btw `  G ) Y )  /\  -.  z  =  X  /\  -.  z  =  Y ) )
75 r19.26-3 2690 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A. z  e.  ( P  i^i  M ) ( -.  z  e.  ( X (btw `  G ) Y )  /\  -.  z  =  X  /\  -.  z  =  Y
)  <->  ( A. z  e.  ( P  i^i  M
)  -.  z  e.  ( X (btw `  G ) Y )  /\  A. z  e.  ( P  i^i  M
)  -.  z  =  X  /\  A. z  e.  ( P  i^i  M
)  -.  z  =  Y ) )
768isibg1a 26214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ph  ->  G  e. Ig )
776, 51, 76, 53isig12 26167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ph  ->  M  C_  P )
78 ssid 3210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  M  C_  M
7978jctr 526 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( M 
C_  P  ->  ( M  C_  P  /\  M  C_  M ) )
80 ssin 3404 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( M  C_  P  /\  M  C_  M )  <->  M  C_  ( P  i^i  M ) )
8179, 80sylib 188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( M 
C_  P  ->  M  C_  ( P  i^i  M
) )
8277, 81syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ph  ->  M  C_  ( P  i^i  M ) )
8382sseld 3192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ph  ->  ( z  e.  M  ->  z  e.  ( P  i^i  M ) ) )
8483imim1d 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ph  ->  ( ( z  e.  ( P  i^i  M
)  ->  -.  z  e.  ( X (btw `  G ) Y ) )  ->  ( z  e.  M  ->  -.  z  e.  ( X (btw `  G ) Y ) ) ) )
8584ralimdv2 2636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  ( A. z  e.  ( P  i^i  M
)  -.  z  e.  ( X (btw `  G ) Y )  ->  A. z  e.  M  -.  z  e.  ( X (btw `  G ) Y ) ) )
86 disjr 3509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( X (btw `  G ) Y )  i^i  M )  =  (/) 
<-> 
A. z  e.  M  -.  z  e.  ( X (btw `  G ) Y ) )
8786biimpri 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( A. z  e.  M  -.  z  e.  ( X
(btw `  G ) Y )  ->  (
( X (btw `  G ) Y )  i^i  M )  =  (/) )
8887a1d 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A. z  e.  M  -.  z  e.  ( X
(btw `  G ) Y )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =  (/) ) )
8985, 88syl6 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( A. 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 )  =  (/) ) ) )
9089adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( A. 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 )  =  (/) ) ) )
9190pm2.43b 46 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A. 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 )  =  (/) ) )
92913ad2ant1 976 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( A. z  e.  ( P  i^i  M )  -.  z  e.  ( X (btw `  G
) Y )  /\  A. z  e.  ( P  i^i  M )  -.  z  =  X  /\  A. z  e.  ( P  i^i  M )  -.  z  =  Y )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( X (btw
`  G ) Y )  i^i  M )  =  (/) ) )
9375, 92sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A. 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 )  =  (/) ) )
9474, 93sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A. 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 )  =  (/) ) )
9572, 94sylbi 187 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { 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 )  =  (/) ) )
9671, 95syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( 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 )  =  (/) ) )
9796ex 423 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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 )  =  (/) ) ) )
9870, 97syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( (
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 )  =  (/) ) ) )
9998pm2.43a 45 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( X S Y )  i^i  M
)  =  (/)  ->  (
( X (btw `  G ) Y )  i^i  M )  =  (/) ) )
10099ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( ( X S Y )  i^i  M
)  =  (/)  ->  (
( X (btw `  G ) Y )  i^i  M )  =  (/) ) ) )
10128, 100mpid 37 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( X (btw `  G ) Y )  i^i  M )  =  (/) ) )
102101imp 418 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( X (btw `  G ) Y )  i^i  M )  =  (/) )
10341adantr 451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Z  e.  P )
104 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Y  =/=  Z  <->  -.  Y  =  Z )
105104biimpri 197 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  Y  =  Z  ->  Y  =/=  Z )
1061053ad2ant1 976 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  Y  =/=  Z )
107106adantl 452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  Y  =/=  Z )
1086, 7, 52, 62, 38, 103, 107sgplpte21 26235 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( Y S Z )  =  { z  e.  P  |  ( z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z ) } )
109108ineq1d 3382 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
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 ) )
110 inrab2 3454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { 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 ) }
111109, 110syl6eq 2344 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
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 ) } )
112 eqtr2 2314 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( 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 ) }  =  (/) )
113 rabeq0 3489 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z ) }  =  (/)  <->  A. z  e.  ( P  i^i  M
)  -.  ( z  e.  ( Y (btw
`  G ) Z )  \/  z  =  Y  \/  z  =  Z ) )
114 3ioran 950 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z )  <->  ( -.  z  e.  ( Y (btw `  G
) Z )  /\  -.  z  =  Y  /\  -.  z  =  Z ) )
115114ralbii 2580 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A. z  e.  ( P  i^i  M )  -.  (
z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z )  <->  A. z  e.  ( P  i^i  M
) ( -.  z  e.  ( Y (btw `  G ) Z )  /\  -.  z  =  Y  /\  -.  z  =  Z ) )
116 r19.26-3 2690 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A. z  e.  ( P  i^i  M ) ( -.  z  e.  ( Y (btw `  G ) Z )  /\  -.  z  =  Y  /\  -.  z  =  Z
)  <->  ( A. z  e.  ( P  i^i  M
)  -.  z  e.  ( Y (btw `  G ) Z )  /\  A. z  e.  ( P  i^i  M
)  -.  z  =  Y  /\  A. z  e.  ( P  i^i  M
)  -.  z  =  Z ) )
11783imim1d 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ph  ->  ( ( z  e.  ( P  i^i  M
)  ->  -.  z  e.  ( Y (btw `  G ) Z ) )  ->  ( z  e.  M  ->  -.  z  e.  ( Y (btw `  G ) Z ) ) ) )
118117ralimdv2 2636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  ( A. z  e.  ( P  i^i  M
)  -.  z  e.  ( Y (btw `  G ) Z )  ->  A. z  e.  M  -.  z  e.  ( Y (btw `  G ) Z ) ) )
119 disjr 3509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( Y (btw `  G ) Z )  i^i  M )  =  (/) 
<-> 
A. z  e.  M  -.  z  e.  ( Y (btw `  G ) Z ) )
120119biimpri 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( A. z  e.  M  -.  z  e.  ( Y
(btw `  G ) Z )  ->  (
( Y (btw `  G ) Z )  i^i  M )  =  (/) )
121120a1d 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A. z  e.  M  -.  z  e.  ( Y
(btw `  G ) Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
122118, 121syl6 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( A. z  e.  ( P  i^i  M
)  -.  z  e.  ( Y (btw `  G ) Z )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) ) )
123122adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( A. z  e.  ( P  i^i  M )  -.  z  e.  ( Y (btw `  G ) Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) ) )
124123pm2.43b 46 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A. z  e.  ( P  i^i  M )  -.  z  e.  ( Y (btw `  G ) Z )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
1251243ad2ant1 976 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( A. z  e.  ( P  i^i  M )  -.  z  e.  ( Y (btw `  G
) Z )  /\  A. z  e.  ( P  i^i  M )  -.  z  =  Y  /\  A. z  e.  ( P  i^i  M )  -.  z  =  Z )  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
126116, 125sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A. z  e.  ( P  i^i  M ) ( -.  z  e.  ( Y (btw `  G ) Z )  /\  -.  z  =  Y  /\  -.  z  =  Z
)  ->  ( ( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
127115, 126sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A. z  e.  ( P  i^i  M )  -.  (
z  e.  ( Y (btw `  G ) Z )  \/  z  =  Y  \/  z  =  Z )  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
128113, 127sylbi 187 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { z  e.  ( P  i^i  M )  |  ( z  e.  ( Y (btw `  G
) Z )  \/  z  =  Y  \/  z  =  Z ) }  =  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) )
129112, 128syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( 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 (btw
`  G ) Z )  i^i  M )  =  (/) ) )
130129ex 423 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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 (btw
`  G ) Z )  i^i  M )  =  (/) ) ) )
131111, 130syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( Y S Z )  i^i  M
)  =  (/)  ->  (
( ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  -> 
( ( Y (btw
`  G ) Z )  i^i  M )  =  (/) ) ) )
132131pm2.43a 45 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( ( Y S Z )  i^i  M
)  =  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =  (/) ) )
133132ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( ( Y S Z )  i^i  M
)  =  (/)  ->  (
( Y (btw `  G ) Z )  i^i  M )  =  (/) ) ) )
1341, 133mpid 37 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  ->  (
( Y (btw `  G ) Z )  i^i  M )  =  (/) ) )
135134imp 418 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( Y (btw `  G ) Z )  i^i  M )  =  (/) )
1366, 51, 38, 52, 54, 55, 57, 58, 102, 135bsstr 26231 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( X (btw `  G ) Z )  i^i  M )  =  (/) )
13750, 136syl5eq 2340 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( M  i^i  ( X (btw
`  G ) Z ) )  =  (/) )
138 disj 3508 . . . . . . . . . . . . 13  |-  ( ( M  i^i  ( X (btw `  G ) Z ) )  =  (/) 
<-> 
A. x  e.  M  -.  x  e.  ( X (btw `  G ) Z ) )
139137, 138sylib 188 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  M  -.  x  e.  ( X (btw `  G ) Z ) )
140 risset 2603 . . . . . . . . . . . . . . 15  |-  ( X  e.  M  <->  E. x  e.  M  x  =  X )
14116, 140sylnib 295 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  E. x  e.  M  x  =  X )
142141adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  -.  E. x  e.  M  x  =  X )
143 ralnex 2566 . . . . . . . . . . . . 13  |-  ( A. x  e.  M  -.  x  =  X  <->  -.  E. x  e.  M  x  =  X )
144142, 143sylibr 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  M  -.  x  =  X )
145 eldifn 3312 . . . . . . . . . . . . . . . 16  |-  ( Z  e.  ( P  \  M )  ->  -.  Z  e.  M )
14639, 145syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  Z  e.  M
)
147146adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  -.  Z  e.  M )
148 risset 2603 . . . . . . . . . . . . . 14  |-  ( Z  e.  M  <->  E. x  e.  M  x  =  Z )
149147, 148sylnib 295 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  -.  E. x  e.  M  x  =  Z )
150 ralnex 2566 . . . . . . . . . . . . 13  |-  ( A. x  e.  M  -.  x  =  Z  <->  -.  E. x  e.  M  x  =  Z )
151149, 150sylibr 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  M  -.  x  =  Z )
152 r19.26-3 2690 . . . . . . . . . . . 12  |-  ( 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 ) )
153139, 144, 151, 152syl3anbrc 1136 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  M  ( -.  x  e.  ( X
(btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z
) )
154 inss2 3403 . . . . . . . . . . . . . . . 16  |-  ( P  i^i  M )  C_  M
155154a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( P  i^i  M
)  C_  M )
156155sseld 3192 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( P  i^i  M )  ->  x  e.  M
) )
157156imim1d 69 . . . . . . . . . . . . 13  |-  ( 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 ) ) ) )
158157adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  (
( 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 ) ) ) )
159158ralimdv2 2636 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( 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 ) ) )
160153, 159mpd 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  ( P  i^i  M
) ( -.  x  e.  ( X (btw `  G ) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) )
161 3ioran 950 . . . . . . . . . . . 12  |-  ( -.  ( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z )  <->  ( -.  x  e.  ( X (btw `  G
) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) )
162161a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( -.  ( x  e.  ( X (btw `  G
) Z )  \/  x  =  X  \/  x  =  Z )  <->  ( -.  x  e.  ( X (btw `  G
) Z )  /\  -.  x  =  X  /\  -.  x  =  Z ) ) )
163162ralbidv 2576 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  ( 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 ) ) )
164160, 163mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y ) )  ->  A. x  e.  ( P  i^i  M
)  -.  ( x  e.  ( X (btw
`  G ) Z )  \/  x  =  X  \/  x  =  Z ) )
165164ancoms 439 . . . . . . . 8  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  A. x  e.  ( P  i^i  M )  -.  ( x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) )
166 rabeq0 3489 . . . . . . . 8  |-  ( { 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 ) )
167165, 166sylibr 203 . . . . . . 7  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  { x  e.  ( P  i^i  M
)  |  ( x  e.  ( X (btw
`  G ) Z )  \/  x  =  X  \/  x  =  Z ) }  =  (/) )
16849, 167syl5eq 2340 . . . . . 6  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( { x  e.  P  |  (
x  e.  ( X (btw `  G ) Z )  \/  x  =  X  \/  x  =  Z ) }  i^i  M )  =  (/) )
16948, 168eqtrd 2328 . . . . 5  |-  ( ( ( -.  Y  =  Z  /\  -.  X  =  Z  /\  -.  X  =  Y )  /\  ph )  ->  ( ( X S Z )  i^i 
M )  =  (/) )
1701693exp1 1167 . . . 4  |-  ( -.  Y  =  Z  -> 
( -.  X  =  Z  ->  ( -.  X  =  Y  ->  (
ph  ->  ( ( X S Z )  i^i 
M )  =  (/) ) ) ) )
17135, 170pm2.61i 156 . . 3  |-  ( -.  X  =  Z  -> 
( -.  X  =  Y  ->  ( ph  ->  ( ( X S Z )  i^i  M
)  =  (/) ) ) )
17227, 171pm2.61i 156 . 2  |-  ( -.  X  =  Y  -> 
( ph  ->  ( ( X S Z )  i^i  M )  =  (/) ) )
1735, 172pm2.61i 156 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:  bosser  26270  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-ig2 26164  df-ibg2 26212  df-seg2 26234
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