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Theorem 2spotdisj 25789
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 387 . . . . . . . . . 10  |-  ( b  =  c  ->  (
b  =  c  \/  ( ( A ( V 2SPathOnOt  E ) b )  i^i  ( A ( V 2SPathOnOt  E ) c ) )  =  (/) ) )
21a1d 26 . . . . . . . . 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 25594 . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . 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 25594 . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . 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 721 . . . . . . . . . . . . . . 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 3635 . . . . . . . . . . . . . 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 721 . . . . . . . . . . . . 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 734 . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( 2nd `  t
)  =  b  /\  ( 2nd `  t )  =  c )  -> 
b  =  c )
1312ex 436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( 2nd `  t )  =  b  ->  (
( 2nd `  t
)  =  c  -> 
b  =  c ) )
1413con3rr3 142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( -.  b  =  c  -> 
( ( 2nd `  t
)  =  b  ->  -.  ( 2nd `  t
)  =  c ) )
1514adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( -.  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 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( -.  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 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( 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 1031 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 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 1031 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( 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 432 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )  ->  -.  ( 2nd `  t )  =  c )
2120intn3an3d 1381 . . . . . . . . . . . . . . . . . . . . . . . 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 ) )
22213mix3d 1185 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
2322ex 436 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
2423exlimdvv 1780 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
2524imp 431 . . . . . . . . . . . . . . . . . . . 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 ) ) )
2625alrimivv 1774 . . . . . . . . . . . . . . . . . . 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 ) ) )
27 2nexaln 1702 . . . . . . . . . . . . . . . . . . . 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 ) ) )
28 3ianor 1002 . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
2928bicomi 206 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
30292albii 1692 . . . . . . . . . . . . . . . . . . . 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 ) ) )
3127, 30bitr4i 256 . . . . . . . . . . . . . . . . . . 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 ) ) )
3226, 31sylibr 216 . . . . . . . . . . . . . . . . . 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 ) ) )
3332intnand 927 . . . . . . . . . . . . . . . . 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 ) ) ) )
34 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  t  ->  ( 1st `  u )  =  ( 1st `  t
) )
3534fveq2d 5869 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 1st `  ( 1st `  u
) )  =  ( 1st `  ( 1st `  t ) ) )
3635eqeq1d 2453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 1st `  ( 1st `  u ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
3734fveq2d 5869 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  ( 1st `  u
) )  =  ( 2nd `  ( 1st `  t ) ) )
3837eqeq1d 2453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  ( 1st `  u ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 ) ) )
39 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  u )  =  ( 2nd `  t
) )
4039eqeq1d 2453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  u
)  =  c  <->  ( 2nd `  t )  =  c ) )
4136, 38, 403anbi123d 1339 . . . . . . . . . . . . . . . . . . . 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 ) ) )
42413anbi3d 1345 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
43422exbidv 1770 . . . . . . . . . . . . . . . . . 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 ) ) ) )
4443elrab 3196 . . . . . . . . . . . . . . . . 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 ) ) ) )
4533, 44sylnibr 307 . . . . . . . . . . . . . . . 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 ) ) } )
4645ex 436 . . . . . . . . . . . . . . 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 ) ) } ) )
4746ralrimiva 2802 . . . . . . . . . . . . . 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 ) ) } ) )
48 fveq2 5865 . . . . . . . . . . . . . . . . . . . 20  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
4948fveq2d 5869 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  t ) ) )
5049eqeq1d 2453 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 1st `  ( 1st `  s ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
5148fveq2d 5869 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
5251eqeq1d 2453 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  ( q ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 ) ) )
53 fveq2 5865 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  s )  =  ( 2nd `  t
) )
5453eqeq1d 2453 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  s
)  =  b  <->  ( 2nd `  t )  =  b ) )
5550, 52, 543anbi123d 1339 . . . . . . . . . . . . . . . . 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 ) ) )
56553anbi3d 1345 . . . . . . . . . . . . . . . 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 ) ) ) )
57562exbidv 1770 . . . . . . . . . . . . . . 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 ) ) ) )
5857ralrab 3200 . . . . . . . . . . . . . 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 ) ) } ) )
5947, 58sylibr 216 . . . . . . . . . . . . 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 ) ) } )
60 disj 3805 . . . . . . . . . . . . 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 ) ) } )
6159, 60sylibr 216 . . . . . . . . . . . 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 ) ) } )  =  (/) )
6211, 61eqtrd 2485 . . . . . . . . . . 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 ) )  =  (/) )
6362olcd 395 . . . . . . . . . 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 ) )  =  (/) ) )
6463ex 436 . . . . . . . . 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 ) )  =  (/) ) ) )
652, 64pm2.61i 168 . . . . . . . 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 ) )  =  (/) ) )
6665ex 436 . . . . . . 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 ) )  =  (/) ) ) )
6766ralrimiva 2802 . . . . . 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 ) )  =  (/) ) ) )
68 raldifb 3573 . . . . . 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 ) )  =  (/) ) )
6967, 68sylib 200 . . . . 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 ) )  =  (/) ) )
7069ex 436 . . . 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 ) )  =  (/) ) ) )
7170ralrimiva 2802 . . 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 ) )  =  (/) ) ) )
72 raldifb 3573 . . 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 ) )  =  (/) ) )
7371, 72sylib 200 . 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 ) )  =  (/) ) )
74 oveq2 6298 . . 3  |-  ( b  =  c  ->  ( A ( V 2SPathOnOt  E ) b )  =  ( A ( V 2SPathOnOt  E ) c ) )
7574disjor 4387 . 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 ) )  =  (/) ) )
7673, 75sylibr 216 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 370    /\ wa 371    \/ w3o 984    /\ w3a 985   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887    e/ wnel 2623   A.wral 2737   {crab 2741    \ cdif 3401    i^i cin 3403   (/)c0 3731   {csn 3968  Disj wdisj 4373   class class class wbr 4402    X. cxp 4832   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   1c1 9540   2c2 10659   #chash 12515   SPathOn cspthon 25233   2SPathOnOt c2pthonot 25585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-2spthonot 25588
This theorem is referenced by:  frghash2spot  25791
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