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Theorem isibg1a3a 26225
Description: If  Y is between  X and 
Z, then  X and  Y, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg1a3a.1  |-  P  =  (PPoints `  G )
isibg1a3a.2  |-  B  =  (btw `  G )
isibg1a3a.3  |-  ( ph  ->  G  e. Ibg )
isibg1a3a.4  |-  ( ph  ->  X  e.  P )
isibg1a3a.5  |-  ( ph  ->  Y  e.  P )
isibg1a3a.6  |-  ( ph  ->  Z  e.  P )
isibg1a3a.7  |-  ( ph  ->  Y  e.  ( X B Z ) )
Assertion
Ref Expression
isibg1a3a  |-  ( ph  ->  X  =/=  Y )

Proof of Theorem isibg1a3a
Dummy variables  u  t  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg1a3a.3 . 2  |-  ( ph  ->  G  e. Ibg )
2 isibg1a3a.1 . . . 4  |-  P  =  (PPoints `  G )
3 eqid 2296 . . . 4  |-  (PLines `  G )  =  (PLines `  G )
4 isibg1a3a.2 . . . 4  |-  B  =  (btw `  G )
5 eqid 2296 . . . 4  |-  (coln `  G )  =  (coln `  G )
62, 3, 4, 5isibg2 26213 . . 3  |-  ( G  e. Ibg 
<->  ( G  e. Ig  /\  A. z  e.  P  A. y  e.  P  (
( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  (
( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) ) ) )
7 isibg1a3a.4 . . . . . . 7  |-  ( ph  ->  X  e.  P )
8 isibg1a3a.5 . . . . . . 7  |-  ( ph  ->  Y  e.  P )
92, 3isibg2aalem1 26216 . . . . . . . . 9  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( A. z  e.  P  A. y  e.  P  ( ( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  ( ( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  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. x  e.  P  ( ( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) )  ->  (
( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  ->  ( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t
)  =  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) ) ) ) ) ) ) ) )
10 isibg1a3a.6 . . . . . . . . . . . 12  |-  ( ph  ->  Z  e.  P )
11 isibg2aalem2 26217 . . . . . . . . . . . . . 14  |-  ( Z  e.  P  ->  ( A. x  e.  P  ( ( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) )  -> 
( ( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  -> 
( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t )  =  (/) )  ->  ( ( X B x )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  ->  ( ( X B x )  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 isibg1a3a.7 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Y  e.  ( X B Z ) )
13 simp31 991 . . . . . . . . . . . . . . . 16  |-  ( ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  X  =/=  Y )
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  ->  X  =/= 
Y ) )
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  ->  X  =/= 
Y ) )
1611, 15syl6 29 . . . . . . . . . . . . 13  |-  ( Z  e.  P  ->  ( A. x  e.  P  ( ( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) )  -> 
( ( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  -> 
( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t )  =  (/) )  ->  ( ( X B x )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  ->  ( ( X B x )  i^i  t )  =  (/) ) ) ) ) )  ->  ( ph  ->  X  =/=  Y ) ) )
1716com23 72 . . . . . . . . . . . 12  |-  ( Z  e.  P  ->  ( ph  ->  ( A. x  e.  P  ( (
( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) )  ->  (
( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  ->  ( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t
)  =  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) ) ) ) ) )  ->  X  =/=  Y ) ) )
1810, 17mpcom 32 . . . . . . . . . . 11  |-  ( ph  ->  ( A. x  e.  P  ( ( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) )  -> 
( ( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  -> 
( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) ) )  /\  A. t  e.  (PLines `  G ) ( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t )  ->  (
( ( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t )  =  (/) )  ->  ( ( X B x )  i^i  t )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  ->  ( ( X B x )  i^i  t )  =  (/) ) ) ) ) )  ->  X  =/=  Y ) )
1918com12 27 . . . . . . . . . 10  |-  ( A. x  e.  P  (
( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) )  ->  (
( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  ->  ( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t
)  =  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) ) ) ) ) )  -> 
( ph  ->  X  =/= 
Y ) )
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. x  e.  P  (
( ( { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x
) )  ->  (
( X  e.  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e.  ( X B x )  /\  x  e/  ( X B Y ) )  \/  ( X  e/  ( Y B x )  /\  Y  e/  ( X B x )  /\  x  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B x )  ->  ( Y  e.  ( x B X )  /\  { X ,  Y ,  x }  e.  (coln `  G )  /\  ( X  =/=  Y  /\  X  =/=  x  /\  Y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( X  e/  t  /\  Y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( X B Y )  i^i  t )  =  (/)  /\  ( ( Y B x )  i^i  t
)  =  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  t )  =/=  (/)  /\  ( ( Y B x )  i^i  t )  =/=  (/) )  -> 
( ( X B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ph  ->  X  =/=  Y ) )
219, 20syl6 29 . . . . . . . 8  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( A. z  e.  P  A. y  e.  P  ( ( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  ( ( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ph  ->  X  =/=  Y ) ) )
2221com23 72 . . . . . . 7  |-  ( ( X  e.  P  /\  Y  e.  P )  ->  ( ph  ->  ( A. z  e.  P  A. y  e.  P  ( ( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  ( ( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  X  =/=  Y
) ) )
237, 8, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ph  ->  ( A. z  e.  P  A. y  e.  P  ( ( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  ( ( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  X  =/=  Y
) ) )
2423pm2.43i 43 . . . . 5  |-  ( ph  ->  ( A. z  e.  P  A. y  e.  P  ( ( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  ( ( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  X  =/=  Y
) )
2524com12 27 . . . 4  |-  ( A. z  e.  P  A. y  e.  P  (
( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  (
( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) )  ->  ( ph  ->  X  =/=  Y ) )
2625adantl 452 . . 3  |-  ( ( G  e. Ig  /\  A. z  e.  P  A. y  e.  P  (
( z  =/=  y  ->  E. w  e.  P  E. v  e.  P  E. u  e.  P  ( z  e.  ( w B y )  /\  v  e.  ( z B y )  /\  y  e.  ( z B u ) ) )  /\  A. x  e.  P  (
( ( { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) )  -> 
( ( z  e.  ( y B x )  /\  y  e/  ( z B x )  /\  x  e/  ( z B y ) )  \/  (
z  e/  ( y B x )  /\  y  e.  ( z B x )  /\  x  e/  ( z B y ) )  \/  ( z  e/  (
y B x )  /\  y  e/  (
z B x )  /\  x  e.  ( z B y ) ) ) )  /\  ( ( y  e.  ( z B x )  ->  ( y  e.  ( x B z )  /\  { z ,  y ,  x }  e.  (coln `  G
)  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z B y )  i^i  t )  =  (/)  /\  ( ( y B x )  i^i  t
)  =  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) )  /\  ( ( ( ( z B y )  i^i  t )  =/=  (/)  /\  ( ( y B x )  i^i  t )  =/=  (/) )  -> 
( ( z B x )  i^i  t
)  =  (/) ) ) ) ) ) ) )  ->  ( ph  ->  X  =/=  Y ) )
276, 26sylbi 187 . 2  |-  ( G  e. Ibg  ->  ( ph  ->  X  =/=  Y ) )
281, 27mpcom 32 1  |-  ( ph  ->  X  =/=  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556   E.wrex 2557    i^i cin 3164   (/)c0 3468   {ctp 3655   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163  colnccol 26193  btwcbtw 26209  Ibgcibg 26210
This theorem is referenced by:  isibg1a8  26230
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-ibg2 26212
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