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Theorem 4cycl4dv 25388
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 25069 . . . . 5  |-  ( V USGrph  E  ->  Fun  E )
2 4pos 10702 . . . . . . . . . . . . . . 15  |-  0  <  4
3 breq2 4405 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
0  <  ( # `  F
)  <->  0  <  4
) )
42, 3mpbiri 237 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  0  <  ( # `  F
) )
5 0nn0 10881 . . . . . . . . . . . . . 14  |-  0  e.  NN0
64, 5jctil 540 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
0  e.  NN0  /\  0  <  ( # `  F
) ) )
7 nvnencycllem 25364 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
9 1lt4 10778 . . . . . . . . . . . . . . 15  |-  1  <  4
10 breq2 4405 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
1  <  ( # `  F
)  <->  1  <  4
) )
119, 10mpbiri 237 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  1  <  ( # `  F
) )
12 1nn0 10882 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1311, 12jctil 540 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
1  e.  NN0  /\  1  <  ( # `  F
) ) )
14 nvnencycllem 25364 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
1 ) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
168, 15anim12d 566 . . . . . . . . . . 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 10777 . . . . . . . . . . . . . . 15  |-  2  <  4
18 breq2 4405 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
2  <  ( # `  F
)  <->  2  <  4
) )
1917, 18mpbiri 237 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  2  <  ( # `  F
) )
20 2nn0 10883 . . . . . . . . . . . . . 14  |-  2  e.  NN0
2119, 20jctil 540 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
2  e.  NN0  /\  2  <  ( # `  F
) ) )
22 nvnencycllem 25364 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
2 ) )  =  { C ,  D }  ->  { C ,  D }  e.  ran  E ) )
24 3lt4 10776 . . . . . . . . . . . . . . 15  |-  3  <  4
25 breq2 4405 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
3  <  ( # `  F
)  <->  3  <  4
) )
2624, 25mpbiri 237 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  3  <  ( # `  F
) )
27 3nn0 10884 . . . . . . . . . . . . . 14  |-  3  e.  NN0
2826, 27jctil 540 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
3  e.  NN0  /\  3  <  ( # `  F
) ) )
29 nvnencycllem 25364 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
3 ) )  =  { D ,  A }  ->  { D ,  A }  e.  ran  E ) )
3123, 30anim12d 566 . . . . . . . . . . 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 566 . . . . . . . . . 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 436 . . . . . . . . 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 437 . . . . . . . 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 431 . . . . . 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 1026 . . . . 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 472 . . . 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 431 . . 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 459 . . . 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 25101 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  =/=  B )
4241ex 436 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  A  =/=  B
) )
4342ad2antrr 731 . . . . . . . . 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 731 . . . . . . 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 431 . . . . . 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 4050 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
4847eqeq2d 2460 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { C ,  B }
) )
49 prcom 4049 . . . . . . . . . . . . . . 15  |-  { B ,  C }  =  { C ,  B }
5049eqeq2i 2462 . . . . . . . . . . . . . 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 716 . . . . . . . . . . . 12  |-  ( A  =  C  ->  (
( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( F `  0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } ) ) )
5352adantr 467 . . . . . . . . . . 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 2471 . . . . . . . . . . . . 13  |-  ( ( ( E `  ( F `  0 )
)  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
55 usgraf1 25080 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
56 wrdf 12673 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
57 oveq2 6296 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  4  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 4 ) )
5857feq2d 5713 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 4 ) --> dom  E ) )
59 4nn 10766 . . . . . . . . . . . . . . . . . . . . 21  |-  4  e.  NN
60 lbfzo0 11952 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  ( 0..^ 4 )  <->  4  e.  NN )
6159, 60mpbir 213 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  ( 0..^ 4 )
62 ffvelrn 6018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  0  e.  (
0..^ 4 ) )  ->  ( F ` 
0 )  e.  dom  E )
6361, 62mpan2 676 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  0 )  e.  dom  E )
64 elfzo0 11953 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  ( 0..^ 4 )  <->  ( 1  e. 
NN0  /\  4  e.  NN  /\  1  <  4
) )
6512, 59, 9, 64mpbir3an 1189 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  ( 0..^ 4 )
66 ffvelrn 6018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  1  e.  (
0..^ 4 ) )  ->  ( F ` 
1 )  e.  dom  E )
6765, 66mpan2 676 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  1 )  e.  dom  E )
6863, 67jca 535 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) )
6958, 68syl6bi 232 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) ) )
7056, 69mpan9 472 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 0 )  e. 
dom  E  /\  ( F `  1 )  e.  dom  E ) )
71703adant2 1026 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  e.  dom  E  /\  ( F ` 
1 )  e.  dom  E ) )
72 f1veqaeq 6159 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 5586 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  <->  ( F : ( 0..^ (
# `  F )
) --> dom  E  /\  Fun  `' F ) )
75 f1eq2 5773 . . . . . . . . . . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  -> 
0  =  1 ) )
79 ax-1ne0 9605 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  =/=  0
8079nesymi 2680 . . . . . . . . . . . . . . . . . . . . . . 23  |-  -.  0  =  1
8180pm2.21i 135 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  1  ->  A  =/=  C )
8278, 81syl6 34 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) )
8376, 82syl6bi 232 . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 0 )  =  ( F `  1
)  ->  A  =/=  C ) ) )
8685ex 436 . . . . . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) )
8988adantl 468 . . . . . . . . . . . . . 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 468 . . . . . . . . . . 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 219 . . . . . . . . . 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 470 . . . . . . . . 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 607 . . . . . . . 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 2706 . . . . . . 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 468 . . . . . 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 25101 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  D  =/=  A )
10099necomd 2678 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  A  =/=  D )
101100ex 436 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { D ,  A }  e.  ran  E  ->  A  =/=  D
) )
102101ad2antrr 731 . . . . . . . . . 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 468 . . . . . . . 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 468 . . . . . . 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 431 . . . . . 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 1187 . . . . 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 25101 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  B  =/=  C )
109108ex 436 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  B  =/=  C
) )
110109ad2antrr 731 . . . . . . . . . 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 468 . . . . . . . 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 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 } ) ) )  ->  B  =/=  C
) )
114113imp 431 . . . . . 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 4049 . . . . . . . . . . . . . . . . . . . . 21  |-  { D ,  A }  =  { A ,  D }
116115eqeq2i 2462 . . . . . . . . . . . . . . . . . . . 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 4051 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  D  ->  { A ,  B }  =  { A ,  D }
)
119118eqeq2d 2460 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  D  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { A ,  D }
) )
120117, 119anbi12d 716 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  D  ->  (
( ( E `  ( F `  3 ) )  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  <->  ( ( E `  ( F `  3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } ) ) )
121120adantr 467 . . . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E `  ( F `  3 )
)  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } )  ->  ( E `  ( F `  3 ) )  =  ( E `  ( F `  0 ) ) )
123 elfzo0 11953 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 3  e.  ( 0..^ 4 )  <->  ( 3  e. 
NN0  /\  4  e.  NN  /\  3  <  4
) )
12427, 59, 24, 123mpbir3an 1189 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  3  e.  ( 0..^ 4 )
125 ffvelrn 6018 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  3  e.  (
0..^ 4 ) )  ->  ( F ` 
3 )  e.  dom  E )
126124, 125mpan2 676 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  3 )  e.  dom  E )
127126, 63jca 535 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) )
12858, 127syl6bi 232 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) ) )
12956, 128mpan9 472 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 3 )  e. 
dom  E  /\  ( F `  0 )  e.  dom  E ) )
1301293adant2 1026 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  e.  dom  E  /\  ( F ` 
0 )  e.  dom  E ) )
131 f1veqaeq 6159 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  -> 
3  =  0 ) )
135 3ne0 10701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  3  =/=  0
136135neii 2625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  -.  3  =  0
137136pm2.21i 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 3  =  0  ->  B  =/=  D )
138134, 137syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) )
13976, 138syl6bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 3 )  =  ( F `  0
)  ->  B  =/=  D ) ) )
142141ex 436 . . . . . . . . . . . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) )
145144adantl 468 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 467 . . . . . . . . . . 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 431 . . . . . . . . . 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 607 . . . . . . . 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 2706 . . . . . . 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 468 . . . . . 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 25101 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { C ,  D }  e.  ran  E )  ->  C  =/=  D )
162161ex 436 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { C ,  D }  e.  ran  E  ->  C  =/=  D
) )
163162ad2antrr 731 . . . . . . . . . 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 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 } ) ) )  ->  C  =/=  D
) )
166165adantl 468 . . . . . . 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 431 . . . . . 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 1187 . . . . 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 535 . . . 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 535 . . 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 674 . 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 436 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   {cpr 3969   class class class wbr 4401   `'ccnv 4832   dom cdm 4833   ran crn 4834   Fun wfun 5575   -->wf 5577   -1-1->wf1 5578   ` cfv 5581  (class class class)co 6288   0cc0 9536   1c1 9537    < clt 9672   NNcn 10606   2c2 10656   3c3 10657   4c4 10658   NN0cn0 10866  ..^cfzo 11912   #chash 12512  Word cword 12653   USGrph cusg 25050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661  df-usgra 25053
This theorem is referenced by:  4cycl4dv4e  25389
  Copyright terms: Public domain W3C validator