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Theorem 4cycl4dv 25380
Description: In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
Assertion
Ref Expression
4cycl4dv  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) ) ) )

Proof of Theorem 4cycl4dv
StepHypRef Expression
1 usgrafun 25062 . . . . 5  |-  ( V USGrph  E  ->  Fun  E )
2 4pos 10705 . . . . . . . . . . . . . . 15  |-  0  <  4
3 breq2 4424 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
0  <  ( # `  F
)  <->  0  <  4
) )
42, 3mpbiri 236 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  0  <  ( # `  F
) )
5 0nn0 10884 . . . . . . . . . . . . . 14  |-  0  e.  NN0
64, 5jctil 539 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
0  e.  NN0  /\  0  <  ( # `  F
) ) )
7 nvnencycllem 25356 . . . . . . . . . . . . 13  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( 0  e. 
NN0  /\  0  <  (
# `  F )
) )  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
86, 7sylan2 476 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
9 1lt4 10781 . . . . . . . . . . . . . . 15  |-  1  <  4
10 breq2 4424 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
1  <  ( # `  F
)  <->  1  <  4
) )
119, 10mpbiri 236 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  1  <  ( # `  F
) )
12 1nn0 10885 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1311, 12jctil 539 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
1  e.  NN0  /\  1  <  ( # `  F
) ) )
14 nvnencycllem 25356 . . . . . . . . . . . . 13  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( 1  e. 
NN0  /\  1  <  (
# `  F )
) )  ->  (
( E `  ( F `  1 )
)  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
1513, 14sylan2 476 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
1 ) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
168, 15anim12d 565 . . . . . . . . . . 11  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
17 2lt4 10780 . . . . . . . . . . . . . . 15  |-  2  <  4
18 breq2 4424 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
2  <  ( # `  F
)  <->  2  <  4
) )
1917, 18mpbiri 236 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  2  <  ( # `  F
) )
20 2nn0 10886 . . . . . . . . . . . . . 14  |-  2  e.  NN0
2119, 20jctil 539 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
2  e.  NN0  /\  2  <  ( # `  F
) ) )
22 nvnencycllem 25356 . . . . . . . . . . . . 13  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( 2  e. 
NN0  /\  2  <  (
# `  F )
) )  ->  (
( E `  ( F `  2 )
)  =  { C ,  D }  ->  { C ,  D }  e.  ran  E ) )
2321, 22sylan2 476 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
2 ) )  =  { C ,  D }  ->  { C ,  D }  e.  ran  E ) )
24 3lt4 10779 . . . . . . . . . . . . . . 15  |-  3  <  4
25 breq2 4424 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
3  <  ( # `  F
)  <->  3  <  4
) )
2624, 25mpbiri 236 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  3  <  ( # `  F
) )
27 3nn0 10887 . . . . . . . . . . . . . 14  |-  3  e.  NN0
2826, 27jctil 539 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
3  e.  NN0  /\  3  <  ( # `  F
) ) )
29 nvnencycllem 25356 . . . . . . . . . . . . 13  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( 3  e. 
NN0  /\  3  <  (
# `  F )
) )  ->  (
( E `  ( F `  3 )
)  =  { D ,  A }  ->  { D ,  A }  e.  ran  E ) )
3028, 29sylan2 476 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
3 ) )  =  { D ,  A }  ->  { D ,  A }  e.  ran  E ) )
3123, 30anim12d 565 . . . . . . . . . . 11  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( ( E `  ( F `
 2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } )  ->  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )
3216, 31anim12d 565 . . . . . . . . . 10  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) )
3332ex 435 . . . . . . . . 9  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  -> 
( ( # `  F
)  =  4  -> 
( ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) ) )
3433expcom 436 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( Fun  E  ->  (
( # `  F )  =  4  ->  (
( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) ) ) )
3534com23 81 . . . . . . 7  |-  ( F  e. Word  dom  E  ->  ( ( # `  F
)  =  4  -> 
( Fun  E  ->  ( ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) ) ) )
3635imp 430 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( Fun  E  ->  ( ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) ) )
37363adant2 1024 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( Fun  E  ->  ( ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) ) )
381, 37mpan9 471 . . . 4  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) ) )
3938imp 430 . . 3  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )
40 simpl 458 . . . 4  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) )
41 usgraedgrn 25094 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  =/=  B )
4241ex 435 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  A  =/=  B
) )
4342ad2antrr 730 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( { A ,  B }  e.  ran  E  ->  A  =/=  B
) )
4443com12 32 . . . . . . . 8  |-  ( { A ,  B }  e.  ran  E  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  B
) )
4544ad2antrr 730 . . . . . . 7  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  /\  ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  B
) )
4645imp 430 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  A  =/=  B )
47 preq1 4076 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
4847eqeq2d 2436 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { C ,  B }
) )
49 prcom 4075 . . . . . . . . . . . . . . 15  |-  { B ,  C }  =  { C ,  B }
5049eqeq2i 2440 . . . . . . . . . . . . . 14  |-  ( ( E `  ( F `
 1 ) )  =  { B ,  C }  <->  ( E `  ( F `  1 ) )  =  { C ,  B } )
5150a1i 11 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( E `  ( F `  1 )
)  =  { B ,  C }  <->  ( E `  ( F `  1
) )  =  { C ,  B }
) )
5248, 51anbi12d 715 . . . . . . . . . . . 12  |-  ( A  =  C  ->  (
( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( F `  0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } ) ) )
5352adantr 466 . . . . . . . . . . 11  |-  ( ( A  =  C  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( F `  0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } ) ) )
54 eqtr3 2450 . . . . . . . . . . . . 13  |-  ( ( ( E `  ( F `  0 )
)  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
55 usgraf1 25073 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
56 wrdf 12668 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
57 oveq2 6309 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  4  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 4 ) )
5857feq2d 5729 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 4 ) --> dom  E ) )
59 4nn 10769 . . . . . . . . . . . . . . . . . . . . 21  |-  4  e.  NN
60 lbfzo0 11955 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  ( 0..^ 4 )  <->  4  e.  NN )
6159, 60mpbir 212 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  ( 0..^ 4 )
62 ffvelrn 6031 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  0  e.  (
0..^ 4 ) )  ->  ( F ` 
0 )  e.  dom  E )
6361, 62mpan2 675 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  0 )  e.  dom  E )
64 elfzo0 11956 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  ( 0..^ 4 )  <->  ( 1  e. 
NN0  /\  4  e.  NN  /\  1  <  4
) )
6512, 59, 9, 64mpbir3an 1187 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  ( 0..^ 4 )
66 ffvelrn 6031 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  1  e.  (
0..^ 4 ) )  ->  ( F ` 
1 )  e.  dom  E )
6765, 66mpan2 675 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  1 )  e.  dom  E )
6863, 67jca 534 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) )
6958, 68syl6bi 231 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) ) )
7056, 69mpan9 471 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 0 )  e. 
dom  E  /\  ( F `  1 )  e.  dom  E ) )
71703adant2 1024 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  e.  dom  E  /\  ( F ` 
1 )  e.  dom  E ) )
72 f1veqaeq 6172 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-> ran 
E  /\  ( ( F `  0 )  e.  dom  E  /\  ( F `  1 )  e.  dom  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  ( E `  ( F `  1 )
)  ->  ( F `  0 )  =  ( F `  1
) ) )
7355, 71, 72syl2an 479 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  0 ) )  =  ( E `
 ( F ` 
1 ) )  -> 
( F `  0
)  =  ( F `
 1 ) ) )
74 df-f1 5602 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  <->  ( F : ( 0..^ (
# `  F )
) --> dom  E  /\  Fun  `' F ) )
75 f1eq2 5788 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0..^ ( # `  F
) )  =  ( 0..^ 4 )  -> 
( F : ( 0..^ ( # `  F
) ) -1-1-> dom  E  <->  F : ( 0..^ 4 ) -1-1-> dom  E ) )
7657, 75syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  <->  F :
( 0..^ 4 )
-1-1-> dom  E ) )
77 f1veqaeq 6172 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F : ( 0..^ 4 ) -1-1-> dom  E  /\  ( 0  e.  ( 0..^ 4 )  /\  1  e.  ( 0..^ 4 ) ) )  ->  ( ( F `
 0 )  =  ( F `  1
)  ->  0  = 
1 ) )
7861, 65, 77mpanr12 689 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  -> 
0  =  1 ) )
79 ax-1ne0 9608 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  =/=  0
8079nesymi 2697 . . . . . . . . . . . . . . . . . . . . . . 23  |-  -.  0  =  1
8180pm2.21i 134 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  1  ->  A  =/=  C )
8278, 81syl6 34 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) )
8376, 82syl6bi 231 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) ) )
8483com12 32 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( # `  F )  =  4  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) ) )
8574, 84sylbir 216 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 0 )  =  ( F `  1
)  ->  A  =/=  C ) ) )
8685ex 435 . . . . . . . . . . . . . . . . 17  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) ) ) )
8756, 86syl 17 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) ) ) )
88873imp 1199 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) )
8988adantl 467 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) )
9073, 89syld 45 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  0 ) )  =  ( E `
 ( F ` 
1 ) )  ->  A  =/=  C ) )
9154, 90syl5 33 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } )  ->  A  =/=  C ) )
9291adantl 467 . . . . . . . . . . 11  |-  ( ( A  =  C  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } )  ->  A  =/=  C ) )
9353, 92sylbid 218 . . . . . . . . . 10  |-  ( ( A  =  C  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  ->  A  =/=  C ) )
9493adantrd 469 . . . . . . . . 9  |-  ( ( A  =  C  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  ->  A  =/=  C ) )
9594expimpd 606 . . . . . . . 8  |-  ( A  =  C  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  C
) )
96 ax-1 6 . . . . . . . 8  |-  ( A  =/=  C  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  C
) )
9795, 96pm2.61ine 2737 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  C
)
9897adantl 467 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  A  =/=  C )
99 usgraedgrn 25094 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  D  =/=  A )
10099necomd 2695 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  A  =/=  D )
101100ex 435 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { D ,  A }  e.  ran  E  ->  A  =/=  D
) )
102101ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( { D ,  A }  e.  ran  E  ->  A  =/=  D
) )
103102com12 32 . . . . . . . . 9  |-  ( { D ,  A }  e.  ran  E  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  D
) )
104103adantl 467 . . . . . . . 8  |-  ( ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  -> 
( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  D
) )
105104adantl 467 . . . . . . 7  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  /\  ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  A  =/=  D
) )
106105imp 430 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  A  =/=  D )
10746, 98, 1063jca 1185 . . . . 5  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  ( A  =/=  B  /\  A  =/= 
C  /\  A  =/=  D ) )
108 usgraedgrn 25094 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  B  =/=  C )
109108ex 435 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  B  =/=  C
) )
110109ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( { B ,  C }  e.  ran  E  ->  B  =/=  C
) )
111110com12 32 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  C
) )
112111adantl 467 . . . . . . . 8  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  C
) )
113112adantr 466 . . . . . . 7  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  /\  ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  C
) )
114113imp 430 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  B  =/=  C )
115 prcom 4075 . . . . . . . . . . . . . . . . . . . . 21  |-  { D ,  A }  =  { A ,  D }
116115eqeq2i 2440 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( F `
 3 ) )  =  { D ,  A }  <->  ( E `  ( F `  3 ) )  =  { A ,  D } )
117116a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  D  ->  (
( E `  ( F `  3 )
)  =  { D ,  A }  <->  ( E `  ( F `  3
) )  =  { A ,  D }
) )
118 preq2 4077 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  D  ->  { A ,  B }  =  { A ,  D }
)
119118eqeq2d 2436 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  D  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { A ,  D }
) )
120117, 119anbi12d 715 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  D  ->  (
( ( E `  ( F `  3 ) )  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  <->  ( ( E `  ( F `  3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } ) ) )
121120adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 3 ) )  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  <->  ( ( E `  ( F `  3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } ) ) )
122 eqtr3 2450 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E `  ( F `  3 )
)  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } )  ->  ( E `  ( F `  3 ) )  =  ( E `  ( F `  0 ) ) )
123 elfzo0 11956 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 3  e.  ( 0..^ 4 )  <->  ( 3  e. 
NN0  /\  4  e.  NN  /\  3  <  4
) )
12427, 59, 24, 123mpbir3an 1187 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  3  e.  ( 0..^ 4 )
125 ffvelrn 6031 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  3  e.  (
0..^ 4 ) )  ->  ( F ` 
3 )  e.  dom  E )
126124, 125mpan2 675 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  3 )  e.  dom  E )
127126, 63jca 534 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) )
12858, 127syl6bi 231 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) ) )
12956, 128mpan9 471 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 3 )  e. 
dom  E  /\  ( F `  0 )  e.  dom  E ) )
1301293adant2 1024 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  e.  dom  E  /\  ( F ` 
0 )  e.  dom  E ) )
131 f1veqaeq 6172 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E : dom  E -1-1-> ran 
E  /\  ( ( F `  3 )  e.  dom  E  /\  ( F `  0 )  e.  dom  E ) )  ->  ( ( E `
 ( F ` 
3 ) )  =  ( E `  ( F `  0 )
)  ->  ( F `  3 )  =  ( F `  0
) ) )
13255, 130, 131syl2an 479 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  3 ) )  =  ( E `
 ( F ` 
0 ) )  -> 
( F `  3
)  =  ( F `
 0 ) ) )
133 f1veqaeq 6172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F : ( 0..^ 4 ) -1-1-> dom  E  /\  ( 3  e.  ( 0..^ 4 )  /\  0  e.  ( 0..^ 4 ) ) )  ->  ( ( F `
 3 )  =  ( F `  0
)  ->  3  = 
0 ) )
134124, 61, 133mpanr12 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  -> 
3  =  0 ) )
135 3ne0 10704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  3  =/=  0
136135neii 2622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  -.  3  =  0
137136pm2.21i 134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 3  =  0  ->  B  =/=  D )
138134, 137syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) )
13976, 138syl6bi 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) ) )
140139com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( # `  F )  =  4  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) ) )
14174, 140sylbir 216 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 3 )  =  ( F `  0
)  ->  B  =/=  D ) ) )
142141ex 435 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) ) ) )
14356, 142syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F  e. Word  dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) ) ) )
1441433imp 1199 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) )
145144adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) )
146132, 145syld 45 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  3 ) )  =  ( E `
 ( F ` 
0 ) )  ->  B  =/=  D ) )
147122, 146syl5 33 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( E `
 ( F ` 
3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } )  ->  B  =/=  D ) )
148147adantl 467 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } )  ->  B  =/=  D ) )
149121, 148sylbid 218 . . . . . . . . . . . . . . . 16  |-  ( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( E `  ( F `
 3 ) )  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  ->  B  =/=  D ) )
150149com12 32 . . . . . . . . . . . . . . 15  |-  ( ( ( E `  ( F `  3 )
)  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  ->  (
( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) ) )  ->  B  =/=  D ) )
151150ex 435 . . . . . . . . . . . . . 14  |-  ( ( E `  ( F `
 3 ) )  =  { D ,  A }  ->  ( ( E `  ( F `
 0 ) )  =  { A ,  B }  ->  ( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  B  =/=  D
) ) )
152151adantl 467 . . . . . . . . . . . . 13  |-  ( ( ( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } )  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  ->  (
( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) ) )  ->  B  =/=  D ) ) )
153152com12 32 . . . . . . . . . . . 12  |-  ( ( E `  ( F `
 0 ) )  =  { A ,  B }  ->  ( ( ( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } )  ->  (
( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) ) )  ->  B  =/=  D ) ) )
154153adantr 466 . . . . . . . . . . 11  |-  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  ->  (
( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } )  ->  (
( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) ) )  ->  B  =/=  D ) ) )
155154imp 430 . . . . . . . . . 10  |-  ( ( ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) ) )  ->  B  =/=  D ) )
156155com12 32 . . . . . . . . 9  |-  ( ( B  =  D  /\  ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) ) )  ->  ( ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  ->  B  =/=  D ) )
157156expimpd 606 . . . . . . . 8  |-  ( B  =  D  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  D
) )
158 ax-1 6 . . . . . . . 8  |-  ( B  =/=  D  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  D
) )
159157, 158pm2.61ine 2737 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  B  =/=  D
)
160159adantl 467 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  B  =/=  D )
161 usgraedgrn 25094 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { C ,  D }  e.  ran  E )  ->  C  =/=  D )
162161ex 435 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { C ,  D }  e.  ran  E  ->  C  =/=  D
) )
163162ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( { C ,  D }  e.  ran  E  ->  C  =/=  D
) )
164163com12 32 . . . . . . . . 9  |-  ( { C ,  D }  e.  ran  E  ->  (
( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  C  =/=  D
) )
165164adantr 466 . . . . . . . 8  |-  ( ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  -> 
( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  C  =/=  D
) )
166165adantl 467 . . . . . . 7  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  /\  ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `  2 ) )  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  C  =/=  D
) )
167166imp 430 . . . . . 6  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  C  =/=  D )
168114, 160, 1673jca 1185 . . . . 5  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )
169107, 168jca 534 . . . 4  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )
17040, 169jca 534 . . 3  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F
)  =  4 ) )  /\  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) ) )  ->  ( (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) ) )
17139, 170mpancom 673 . 2  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) )  /\  ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) ) )  ->  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) ) )
172171ex 435 1  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( ( E `  ( F `
 0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  /\  (
( E `  ( F `  2 )
)  =  { C ,  D }  /\  ( E `  ( F `  3 ) )  =  { D ,  A } ) )  -> 
( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   {cpr 3998   class class class wbr 4420   `'ccnv 4848   dom cdm 4849   ran crn 4850   Fun wfun 5591   -->wf 5593   -1-1->wf1 5594   ` cfv 5597  (class class class)co 6301   0cc0 9539   1c1 9540    < clt 9675   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   NN0cn0 10869  ..^cfzo 11915   #chash 12514  Word cword 12648   USGrph cusg 25043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12515  df-word 12656  df-usgra 25046
This theorem is referenced by:  4cycl4dv4e  25381
  Copyright terms: Public domain W3C validator