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Theorem 2spotdisj 25781
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 27 . . . . . . . . 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 25586 . . . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . . 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 25586 . . . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . 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 3666 . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . 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 733 . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1029 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1029 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1377 . . . . . . . . . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . . . . . . . . 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 1770 . . . . . . . . . . . . . . . . . . . . 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 1765 . . . . . . . . . . . . . . . . . . 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 1699 . . . . . . . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . . . . . . . . . . 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 1689 . . . . . . . . . . . . . . . . . . . 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 925 . . . . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  t  ->  ( 1st `  u )  =  ( 1st `  t
) )
3534fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 1st `  ( 1st `  u
) )  =  ( 1st `  ( 1st `  t ) ) )
3635eqeq1d 2425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 1st `  ( 1st `  u ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
3734fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  ( 1st `  u
) )  =  ( 2nd `  ( 1st `  t ) ) )
3837eqeq1d 2425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  ( 1st `  u ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 ) ) )
39 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  t  ->  ( 2nd `  u )  =  ( 2nd `  t
) )
4039eqeq1d 2425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  t  ->  (
( 2nd `  u
)  =  c  <->  ( 2nd `  t )  =  c ) )
4136, 38, 403anbi123d 1336 . . . . . . . . . . . . . . . . . . . 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 1342 . . . . . . . . . . . . . . . . . . 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 1761 . . . . . . . . . . . . . . . . . 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 3230 . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . . . . 20  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
4948fveq2d 5883 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  t ) ) )
5049eqeq1d 2425 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 1st `  ( 1st `  s ) )  =  A  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
5148fveq2d 5883 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
5251eqeq1d 2425 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  ( q ` 
1 )  <->  ( 2nd `  ( 1st `  t
) )  =  ( q `  1 ) ) )
53 fveq2 5879 . . . . . . . . . . . . . . . . . . 19  |-  ( s  =  t  ->  ( 2nd `  s )  =  ( 2nd `  t
) )
5453eqeq1d 2425 . . . . . . . . . . . . . . . . . 18  |-  ( s  =  t  ->  (
( 2nd `  s
)  =  b  <->  ( 2nd `  t )  =  b ) )
5550, 52, 543anbi123d 1336 . . . . . . . . . . . . . . . . 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 1342 . . . . . . . . . . . . . . . 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 1761 . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . 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 3834 . . . . . . . . . . . . 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 2464 . . . . . . . . . . 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 2840 . . . . . 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 3606 . . . . . 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 2840 . . 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 3606 . . 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 6311 . . 3  |-  ( b  =  c  ->  ( A ( V 2SPathOnOt  E ) b )  =  ( A ( V 2SPathOnOt  E ) c ) )
7574disjor 4406 . 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 982    /\ w3a 983   A.wal 1436    = wceq 1438   E.wex 1660    e. wcel 1869    e/ wnel 2620   A.wral 2776   {crab 2780    \ cdif 3434    i^i cin 3436   (/)c0 3762   {csn 3997  Disj wdisj 4392   class class class wbr 4421    X. cxp 4849   ` cfv 5599  (class class class)co 6303   1stc1st 6803   2ndc2nd 6804   1c1 9542   2c2 10661   #chash 12516   SPathOn cspthon 25225   2SPathOnOt c2pthonot 25577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-disj 4393  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-2spthonot 25580
This theorem is referenced by:  frghash2spot  25783
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