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Theorem 2spotdisj 24766
Description: All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
Assertion
Ref Expression
2spotdisj  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  -> Disj  b  e.  ( V  \  { A } ) ( A ( V 2SPathOnOt  E )
b ) )
Distinct variable groups:    A, b    E, b    V, b    X, b    Y, b

Proof of Theorem 2spotdisj
Dummy variables  c 
f  g  p  q  s  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 385 . . . . . . . . . 10  |-  ( b  =  c  ->  (
b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
21a1d 25 . . . . . . . . 9  |-  ( b  =  c  ->  (
( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } )  -> 
( b  =  c  \/  ( ( A ( V 2SPathOnOt  E )
b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
3 2spthonot 24570 . . . . . . . . . . . . . . . . 17  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  b  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) b )  =  { s  e.  ( ( V  X.  V )  X.  V
)  |  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) } )
43anassrs 648 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  ->  ( A ( V 2SPathOnOt  E ) b )  =  {
s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) } )
54adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  c  e.  V )  ->  ( A ( V 2SPathOnOt  E ) b )  =  {
s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) } )
6 2spthonot 24570 . . . . . . . . . . . . . . . . 17  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  c  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) c )  =  { u  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
76anassrs 648 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  c  e.  V )  ->  ( A ( V 2SPathOnOt  E ) c )  =  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
87adantlr 714 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  c  e.  V )  ->  ( A ( V 2SPathOnOt  E ) c )  =  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
95, 8ineq12d 3701 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  c  e.  V )  ->  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  ( { s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  i^i  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
109adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  ->  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  ( { s  e.  ( ( V  X.  V )  X.  V
)  |  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  i^i  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
1110ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  ( ( A ( V 2SPathOnOt  E )
b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  ( { s  e.  ( ( V  X.  V )  X.  V
)  |  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  i^i  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
12 eqtr2 2494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( 2nd `  t
)  =  b  /\  ( 2nd `  t )  =  c )  -> 
b  =  c )
1312ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( 2nd `  t )  =  b  ->  (
( 2nd `  t
)  =  c  -> 
b  =  c ) )
1413con3rr3 136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( -.  b  =  c  -> 
( ( 2nd `  t
)  =  b  ->  -.  ( 2nd `  t
)  =  c ) )
1514adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  ( ( 2nd `  t )  =  b  ->  -.  ( 2nd `  t )  =  c ) )
1615adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  ( ( 2nd `  t )  =  b  ->  -.  ( 2nd `  t )  =  c ) )
1716com12 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( 2nd `  t )  =  b  ->  (
( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  -.  ( 2nd `  t )  =  c ) )
18173ad2ant3 1019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b )  ->  ( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  -.  ( 2nd `  t )  =  c ) )
19183ad2ant3 1019 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) )  -> 
( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  -.  ( 2nd `  t )  =  c ) )
2019impcom 430 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) ) )  ->  -.  ( 2nd `  t )  =  c )
21 3mix3 1167 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( -.  ( 2nd `  t
)  =  c  -> 
( -.  ( 1st `  ( 1st `  t
) )  =  A  \/  -.  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  \/  -.  ( 2nd `  t )  =  c ) )
2220, 21syl 16 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) ) )  ->  ( -.  ( 1st `  ( 1st `  t
) )  =  A  \/  -.  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  \/  -.  ( 2nd `  t )  =  c ) )
23 3ianor 990 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c )  <-> 
( -.  ( 1st `  ( 1st `  t
) )  =  A  \/  -.  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  \/  -.  ( 2nd `  t )  =  c ) )
2422, 23sylibr 212 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) ) )  ->  -.  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  c ) )
25 3mix3 1167 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c )  ->  ( -.  f
( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
2624, 25syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) ) )  ->  ( -.  f
( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
2726ex 434 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  ( (
g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) )  -> 
( -.  f ( A ( V SPathOn  E
) c ) p  \/  -.  ( # `  f )  =  2  \/  -.  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
2827exlimdvv 1701 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  ( E. g E. q ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) )  -> 
( -.  f ( A ( V SPathOn  E
) c ) p  \/  -.  ( # `  f )  =  2  \/  -.  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
2928imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )  ->  ( -.  f ( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
3029alrimivv 1696 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )  ->  A. f A. p ( -.  f
( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
31 2nexaln 1631 . . . . . . . . . . . . . . . . . . . 20  |-  ( -. 
E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) )  <->  A. f A. p  -.  ( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
32 3ianor 990 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  ( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) )  <->  ( -.  f
( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
3332bicomi 202 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  f ( A ( V SPathOn  E )
c ) p  \/ 
-.  ( # `  f
)  =  2  \/ 
-.  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  c ) )  <->  -.  (
f ( A ( V SPathOn  E ) c ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  c ) ) )
34332albii 1621 . . . . . . . . . . . . . . . . . . . 20  |-  ( A. f A. p ( -.  f ( A ( V SPathOn  E ) c ) p  \/  -.  ( # `
 f )  =  2  \/  -.  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) )  <->  A. f A. p  -.  ( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
3531, 34bitr4i 252 . . . . . . . . . . . . . . . . . . 19  |-  ( -. 
E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) )  <->  A. f A. p
( -.  f ( A ( V SPathOn  E
) c ) p  \/  -.  ( # `  f )  =  2  \/  -.  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
3630, 35sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )  ->  -.  E. f E. p ( f ( A ( V SPathOn  E ) c ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  c ) ) )
3736intnand 914 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )  ->  -.  ( t  e.  ( ( V  X.  V
)  X.  V )  /\  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
38 fveq2 5866 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  t  ->  ( 1st `  u )  =  ( 1st `  t
) )
3938fveq2d 5870 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 1st `  ( 1st `  u
) )  =  ( 1st `  ( 1st `  t ) ) )
4039eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 1st `  ( 1st `  u ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
4138fveq2d 5870 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  ( 1st `  u
) )  =  ( 2nd `  ( 1st `  t ) ) )
4241eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  ( 1st `  u ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 ) ) )
43 fveq2 5866 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  u )  =  ( 2nd `  t
) )
4443eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  u
)  =  c  <->  ( 2nd `  t )  =  c ) )
4540, 42, 443anbi123d 1299 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  t  ->  (
( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c )  <-> 
( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) )
46453anbi3d 1305 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  t  ->  (
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) )  <->  ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
47462exbidv 1692 . . . . . . . . . . . . . . . . . 18  |-  ( u  =  t  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) c ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  u
) )  =  A  /\  ( 2nd `  ( 1st `  u ) )  =  ( p ` 
1 )  /\  ( 2nd `  u )  =  c ) )  <->  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
4847elrab 3261 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  { u  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) }  <->  ( t  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  c ) ) ) )
4937, 48sylnibr 305 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( -.  b  =  c  /\  (
( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  /\  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )  ->  -.  t  e.  { u  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
5049ex 434 . . . . . . . . . . . . . . 15  |-  ( ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  /\  t  e.  ( ( V  X.  V
)  X.  V ) )  ->  ( E. g E. q ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( q ` 
1 )  /\  ( 2nd `  t )  =  b ) )  ->  -.  t  e.  { u  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
5150ralrimiva 2878 . . . . . . . . . . . . . 14  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  A. t  e.  ( ( V  X.  V
)  X.  V ) ( E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) )  ->  -.  t  e.  { u  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
52 fveq2 5866 . . . . . . . . . . . . . . . . . . . 20  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
5352fveq2d 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  t ) ) )
5453eqeq1d 2469 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 1st `  ( 1st `  s ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
5552fveq2d 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
5655eqeq1d 2469 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  ( q ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 ) ) )
57 fveq2 5866 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  s )  =  ( 2nd `  t
) )
5857eqeq1d 2469 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  s
)  =  b  <->  ( 2nd `  t )  =  b ) )
5954, 56, 583anbi123d 1299 . . . . . . . . . . . . . . . . 17  |-  ( s  =  t  ->  (
( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b )  <-> 
( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) )
60593anbi3d 1305 . . . . . . . . . . . . . . . 16  |-  ( s  =  t  ->  (
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) )  <->  ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
61602exbidv 1692 . . . . . . . . . . . . . . 15  |-  ( s  =  t  ->  ( E. g E. q ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  s
) )  =  A  /\  ( 2nd `  ( 1st `  s ) )  =  ( q ` 
1 )  /\  ( 2nd `  s )  =  b ) )  <->  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
6261ralrab 3265 . . . . . . . . . . . . . 14  |-  ( A. t  e.  { s  e.  ( ( V  X.  V )  X.  V
)  |  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  -.  t  e.  { u  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) }  <->  A. t  e.  ( ( V  X.  V )  X.  V
) ( E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 )  /\  ( 2nd `  t
)  =  b ) )  ->  -.  t  e.  { u  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } ) )
6351, 62sylibr 212 . . . . . . . . . . . . 13  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  A. t  e.  {
s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  -.  t  e.  { u  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
64 disj 3867 . . . . . . . . . . . . 13  |-  ( ( { s  e.  ( ( V  X.  V
)  X.  V )  |  E. g E. q ( g ( A ( V SPathOn  E
) b ) q  /\  ( # `  g
)  =  2  /\  ( ( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  i^i  {
u  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E )
c ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )  =  (/) 
<-> 
A. t  e.  {
s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q
( g ( A ( V SPathOn  E )
b ) q  /\  ( # `  g )  =  2  /\  (
( 1st `  ( 1st `  s ) )  =  A  /\  ( 2nd `  ( 1st `  s
) )  =  ( q `  1 )  /\  ( 2nd `  s
)  =  b ) ) }  -.  t  e.  { u  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )
6563, 64sylibr 212 . . . . . . . . . . . 12  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  ( { s  e.  ( ( V  X.  V )  X.  V )  |  E. g E. q ( g ( A ( V SPathOn  E ) b ) q  /\  ( # `  g )  =  2  /\  ( ( 1st `  ( 1st `  s
) )  =  A  /\  ( 2nd `  ( 1st `  s ) )  =  ( q ` 
1 )  /\  ( 2nd `  s )  =  b ) ) }  i^i  { u  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) c ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  u ) )  =  A  /\  ( 2nd `  ( 1st `  u
) )  =  ( p `  1 )  /\  ( 2nd `  u
)  =  c ) ) } )  =  (/) )
6611, 65eqtrd 2508 . . . . . . . . . . 11  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  ( ( A ( V 2SPathOnOt  E )
b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) )
6766olcd 393 . . . . . . . . . 10  |-  ( ( -.  b  =  c  /\  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } ) )  ->  ( b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
6867ex 434 . . . . . . . . 9  |-  ( -.  b  =  c  -> 
( ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } )  -> 
( b  =  c  \/  ( ( A ( V 2SPathOnOt  E )
b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
692, 68pm2.61i 164 . . . . . . . 8  |-  ( ( ( ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  /\  c  e/  { A } )  -> 
( b  =  c  \/  ( ( A ( V 2SPathOnOt  E )
b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
7069ex 434 . . . . . . 7  |-  ( ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V
)  /\  b  e.  V )  /\  b  e/  { A } )  /\  c  e.  V
)  ->  ( c  e/  { A }  ->  ( b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
7170ralrimiva 2878 . . . . . 6  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  ->  A. c  e.  V  ( c  e/  { A }  ->  ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
72 raldifb 3644 . . . . . 6  |-  ( A. c  e.  V  (
c  e/  { A }  ->  ( b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )  <->  A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
7371, 72sylib 196 . . . . 5  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  /\  b  e/  { A } )  ->  A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
7473ex 434 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  /\  b  e.  V )  ->  (
b  e/  { A }  ->  A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
7574ralrimiva 2878 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  ->  A. b  e.  V  ( b  e/  { A }  ->  A. c  e.  ( V 
\  { A }
) ( b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) ) )
76 raldifb 3644 . . 3  |-  ( A. b  e.  V  (
b  e/  { A }  ->  A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )  <->  A. b  e.  ( V  \  { A }
) A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
7775, 76sylib 196 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  ->  A. b  e.  ( V  \  { A } ) A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
78 oveq2 6292 . . 3  |-  ( b  =  c  ->  ( A ( V 2SPathOnOt  E ) b )  =  ( A ( V 2SPathOnOt  E ) c ) )
7978disjor 4431 . 2  |-  (Disj  b  e.  ( V  \  { A } ) ( A ( V 2SPathOnOt  E )
b )  <->  A. b  e.  ( V  \  { A } ) A. c  e.  ( V  \  { A } ) ( b  =  c  \/  (
( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
8077, 79sylibr 212 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  A  e.  V )  -> Disj  b  e.  ( V  \  { A } ) ( A ( V 2SPathOnOt  E )
b ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767    e/ wnel 2663   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475   (/)c0 3785   {csn 4027  Disj wdisj 4417   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   1c1 9493   2c2 10585   #chash 12373   SPathOn cspthon 24209   2SPathOnOt c2pthonot 24561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-2spthonot 24564
This theorem is referenced by:  frghash2spot  24768
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