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Theorem isib2g1a1 26116
Description: If  Y is between  X and 
Z, it is between  Z and  X (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg2a.1  |-  P  =  (PPoints `  G )
isibg2a.2  |-  B  =  (btw `  G )
isibg2a.3  |-  ( ph  ->  G  e. Ibg )
isibg2a.4  |-  ( ph  ->  X  e.  P )
isibg2a.5  |-  ( ph  ->  Y  e.  P )
isib2g1a1.1  |-  ( ph  ->  Z  e.  P )
isib2g1a1.2  |-  ( ph  ->  Y  e.  ( X B Z ) )
Assertion
Ref Expression
isib2g1a1  |-  ( ph  ->  Y  e.  ( Z B X ) )

Proof of Theorem isib2g1a1
Dummy variables  x  y  z  u  t 
v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg2a.3 . 2  |-  ( ph  ->  G  e. Ibg )
2 isibg2a.1 . . . 4  |-  P  =  (PPoints `  G )
3 eqid 2283 . . . 4  |-  (PLines `  G )  =  (PLines `  G )
4 isibg2a.2 . . . 4  |-  B  =  (btw `  G )
5 eqid 2283 . . . 4  |-  (coln `  G )  =  (coln `  G )
62, 3, 4, 5isibg2 26110 . . 3  |-  ( G  e. Ibg 
<->  ( G  e. Ig  /\  A. y  e.  P  A. x  e.  P  (
( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  (
( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) ) ) )
7 isibg2a.4 . . . . . . 7  |-  ( ph  ->  X  e.  P )
8 isibg2a.5 . . . . . . 7  |-  ( ph  ->  Y  e.  P )
92, 3isibg2aalem1 26113 . . . . . . . . 9  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( A. y  e.  P  A. x  e.  P  ( ( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  ( ( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ( X  =/=  Y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( X  e.  ( w B Y )  /\  v  e.  ( X B Y )  /\  Y  e.  ( X B u ) ) )  /\  A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  (coln `  G
)  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  -> 
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t )  =  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) ) ) ) ) ) ) ) )
10 isib2g1a1.1 . . . . . . . . . . . 12  |-  ( ph  ->  Z  e.  P )
11 isibg2aalem2 26114 . . . . . . . . . . . . . 14  |-  ( Z  e.  P  ->  ( A. z  e.  P  ( ( ( { X ,  Y , 
z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  -> 
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t )  =  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) ) ) ) ) )  ->  ( (
( { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )  ->  (
( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  Z  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B Z )  i^i  t
)  =  (/) )  -> 
( ( X B Z )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B Z )  i^i  t )  =/=  (/) )  -> 
( ( X B Z )  i^i  t
)  =  (/) ) ) ) ) ) ) )
12 isib2g1a1.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Y  e.  ( X B Z ) )
13 simp1 955 . . . . . . . . . . . . . . . 16  |-  ( ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Y  e.  ( Z B X ) )
1412, 13imim12i 53 . . . . . . . . . . . . . . 15  |-  ( ( Y  e.  ( X B Z )  -> 
( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) ) )  -> 
( ph  ->  Y  e.  ( Z B X ) ) )
1514ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )  ->  (
( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  Z  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B Z )  i^i  t
)  =  (/) )  -> 
( ( X B Z )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B Z )  i^i  t )  =/=  (/) )  -> 
( ( X B Z )  i^i  t
)  =  (/) ) ) ) ) )  -> 
( ph  ->  Y  e.  ( Z B X ) ) )
1611, 15syl6 29 . . . . . . . . . . . . 13  |-  ( Z  e.  P  ->  ( A. z  e.  P  ( ( ( { X ,  Y , 
z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  -> 
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t )  =  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) ) )
1716com23 72 . . . . . . . . . . . 12  |-  ( Z  e.  P  ->  ( ph  ->  ( A. z  e.  P  ( (
( { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) )  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y , 
z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t
)  =  (/) )  -> 
( ( X B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  -> 
( ( X B z )  i^i  t
)  =  (/) ) ) ) ) )  ->  Y  e.  ( Z B X ) ) ) )
1810, 17mpcom 32 . . . . . . . . . . 11  |-  ( ph  ->  ( A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) )  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y , 
z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t
)  =  (/) )  -> 
( ( X B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  -> 
( ( X B z )  i^i  t
)  =  (/) ) ) ) ) )  ->  Y  e.  ( Z B X ) ) )
1918com12 27 . . . . . . . . . 10  |-  ( A. z  e.  P  (
( ( { X ,  Y ,  z }  e.  (coln `  G
)  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  -> 
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t )  =  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) )
2019adantl 452 . . . . . . . . 9  |-  ( ( ( X  =/=  Y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( X  e.  (
w B Y )  /\  v  e.  ( X B Y )  /\  Y  e.  ( X B u ) ) )  /\  A. z  e.  P  (
( ( { X ,  Y ,  z }  e.  (coln `  G
)  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  -> 
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  z  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B z )  i^i  t )  =  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B z )  i^i  t )  =/=  (/) )  ->  ( ( X B z )  i^i  t )  =  (/) ) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) )
219, 20syl6 29 . . . . . . . 8  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( A. y  e.  P  A. x  e.  P  ( ( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  ( ( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) ) )
2221com23 72 . . . . . . 7  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( ph  ->  ( A. y  e.  P  A. x  e.  P  ( ( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  ( ( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  Y  e.  ( Z B X ) ) ) )
237, 8, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ph  ->  ( A. y  e.  P  A. x  e.  P  ( ( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  ( ( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  Y  e.  ( Z B X ) ) ) )
2423pm2.43i 43 . . . . 5  |-  ( ph  ->  ( A. y  e.  P  A. x  e.  P  ( ( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  ( ( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  Y  e.  ( Z B X ) ) )
2524com12 27 . . . 4  |-  ( A. y  e.  P  A. x  e.  P  (
( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  (
( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) )
2625adantl 452 . . 3  |-  ( ( G  e. Ig  /\  A. y  e.  P  A. x  e.  P  (
( y  =/=  x  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( y  e.  ( w B x )  /\  v  e.  ( y B x )  /\  x  e.  ( y B u ) ) )  /\  A. z  e.  P  (
( ( { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) )  -> 
( ( y  e.  ( x B z )  /\  x  e/  ( y B z )  /\  z  e/  ( y B x ) )  \/  (
y  e/  ( x B z )  /\  x  e.  ( y B z )  /\  z  e/  ( y B x ) )  \/  ( y  e/  (
x B z )  /\  x  e/  (
y B z )  /\  z  e.  ( y B x ) ) ) )  /\  ( ( x  e.  ( y B z )  ->  ( x  e.  ( z B y )  /\  { y ,  x ,  z }  e.  (coln `  G )  /\  (
y  =/=  x  /\  y  =/=  z  /\  x  =/=  z ) ) )  /\  A. t  e.  (PLines `  G )
( ( y  e/  t  /\  x  e/  t  /\  z  e/  t
)  ->  ( (
( ( ( y B x )  i^i  t )  =  (/)  /\  ( ( x B z )  i^i  t
)  =  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) )  /\  ( ( ( ( y B x )  i^i  t )  =/=  (/)  /\  ( ( x B z )  i^i  t )  =/=  (/) )  -> 
( ( y B z )  i^i  t
)  =  (/) ) ) ) ) ) ) )  ->  ( ph  ->  Y  e.  ( Z B X ) ) )
276, 26sylbi 187 . 2  |-  ( G  e. Ibg  ->  ( ph  ->  Y  e.  ( Z B X ) ) )
281, 27mpcom 32 1  |-  ( ph  ->  Y  e.  ( Z B X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543   E.wrex 2544    i^i cin 3151   (/)c0 3455   {ctp 3642   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  colnccol 26090  btwcbtw 26106  Ibgcibg 26107
This theorem is referenced by:  xsyysx  26145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-ibg2 26109
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