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Theorem 4cycl4dv 21503
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 21246 . . . . 5  |-  ( V USGrph  E  ->  Fun  E )
2 4pos 10019 . . . . . . . . . . . . . . 15  |-  0  <  4
3 breq2 4158 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
0  <  ( # `  F
)  <->  0  <  4
) )
42, 3mpbiri 225 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  0  <  ( # `  F
) )
5 0nn0 10169 . . . . . . . . . . . . . 14  |-  0  e.  NN0
64, 5jctil 524 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
0  e.  NN0  /\  0  <  ( # `  F
) ) )
7 nvnencycllem 21479 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
9 1lt4 10080 . . . . . . . . . . . . . . 15  |-  1  <  4
10 breq2 4158 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
1  <  ( # `  F
)  <->  1  <  4
) )
119, 10mpbiri 225 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  1  <  ( # `  F
) )
12 1nn0 10170 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1311, 12jctil 524 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
1  e.  NN0  /\  1  <  ( # `  F
) ) )
14 nvnencycllem 21479 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
1 ) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
168, 15anim12d 547 . . . . . . . . . . 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 10079 . . . . . . . . . . . . . . 15  |-  2  <  4
18 breq2 4158 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
2  <  ( # `  F
)  <->  2  <  4
) )
1917, 18mpbiri 225 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  2  <  ( # `  F
) )
20 2nn0 10171 . . . . . . . . . . . . . 14  |-  2  e.  NN0
2119, 20jctil 524 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
2  e.  NN0  /\  2  <  ( # `  F
) ) )
22 nvnencycllem 21479 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
2 ) )  =  { C ,  D }  ->  { C ,  D }  e.  ran  E ) )
24 3lt4 10078 . . . . . . . . . . . . . . 15  |-  3  <  4
25 breq2 4158 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
3  <  ( # `  F
)  <->  3  <  4
) )
2624, 25mpbiri 225 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  3  <  ( # `  F
) )
27 3nn0 10172 . . . . . . . . . . . . . 14  |-  3  e.  NN0
2826, 27jctil 524 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
3  e.  NN0  /\  3  <  ( # `  F
) ) )
29 nvnencycllem 21479 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( # `  F
)  =  4 )  ->  ( ( E `
 ( F ` 
3 ) )  =  { D ,  A }  ->  { D ,  A }  e.  ran  E ) )
3123, 30anim12d 547 . . . . . . . . . . 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 547 . . . . . . . . . 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 424 . . . . . . . . 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 425 . . . . . . . 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 74 . . . . . . 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 419 . . . . . 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 976 . . . . 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 456 . . . 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 419 . . 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 444 . . . 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 21268 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  =/=  B )
4241ex 424 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  A  =/=  B
) )
4342ad2antrr 707 . . . . . . . . 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 29 . . . . . . . 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 707 . . . . . . 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 419 . . . . . 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 3827 . . . . . . . . . . . . . 14  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
4847eqeq2d 2399 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { C ,  B }
) )
49 prcom 3826 . . . . . . . . . . . . . . 15  |-  { B ,  C }  =  { C ,  B }
5049eqeq2i 2398 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . 12  |-  ( A  =  C  ->  (
( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( F `  0 ) )  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } ) ) )
5352adantr 452 . . . . . . . . . . 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 2407 . . . . . . . . . . . . 13  |-  ( ( ( E `  ( F `  0 )
)  =  { C ,  B }  /\  ( E `  ( F `  1 ) )  =  { C ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
55 usgraf1 21251 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
56 wrdf 11661 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
57 oveq2 6029 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  4  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 4 ) )
5857feq2d 5522 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 4 ) --> dom  E ) )
59 4nn 10068 . . . . . . . . . . . . . . . . . . . . 21  |-  4  e.  NN
60 lbfzo0 11101 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  ( 0..^ 4 )  <->  4  e.  NN )
6159, 60mpbir 201 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  ( 0..^ 4 )
62 ffvelrn 5808 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  0  e.  (
0..^ 4 ) )  ->  ( F ` 
0 )  e.  dom  E )
6361, 62mpan2 653 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  0 )  e.  dom  E )
64 elfzo0 11102 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  ( 0..^ 4 )  <->  ( 1  e. 
NN0  /\  4  e.  NN  /\  1  <  4
) )
6512, 59, 9, 64mpbir3an 1136 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  ( 0..^ 4 )
66 ffvelrn 5808 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  1  e.  (
0..^ 4 ) )  ->  ( F ` 
1 )  e.  dom  E )
6765, 66mpan2 653 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  1 )  e.  dom  E )
6863, 67jca 519 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) )
6958, 68syl6bi 220 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  0
)  e.  dom  E  /\  ( F `  1
)  e.  dom  E
) ) )
7056, 69mpan9 456 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 0 )  e. 
dom  E  /\  ( F `  1 )  e.  dom  E ) )
71703adant2 976 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  e.  dom  E  /\  ( F ` 
1 )  e.  dom  E ) )
72 f1veqaeq 5945 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 5400 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  <->  ( F : ( 0..^ (
# `  F )
) --> dom  E  /\  Fun  `' F ) )
75 f1eq2 5576 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0..^ ( # `  F
) )  =  ( 0..^ 4 )  -> 
( F : ( 0..^ ( # `  F
) ) -1-1-> dom  E  <->  F : ( 0..^ 4 ) -1-1-> dom  E ) )
7657, 75syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  <->  F :
( 0..^ 4 )
-1-1-> dom  E ) )
77 f1veqaeq 5945 . . . . . . . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  -> 
0  =  1 ) )
79 ax-1ne0 8993 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  =/=  0
8079necomi 2633 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  =/=  1
81 df-ne 2553 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  =/=  1  <->  -.  0  =  1 )
8280, 81mpbi 200 . . . . . . . . . . . . . . . . . . . . . . 23  |-  -.  0  =  1
8382pm2.21i 125 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  1  ->  A  =/=  C )
8478, 83syl6 31 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) )
8576, 84syl6bi 220 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) ) )
8685com12 29 . . . . . . . . . . . . . . . . . . 19  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( # `  F )  =  4  ->  (
( F `  0
)  =  ( F `
 1 )  ->  A  =/=  C ) ) )
8774, 86sylbir 205 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 0 )  =  ( F `  1
)  ->  A  =/=  C ) ) )
8887ex 424 . . . . . . . . . . . . . . . . 17  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) ) ) )
8956, 88syl 16 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) ) ) )
90893imp 1147 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) )
9190adantl 453 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( F ` 
0 )  =  ( F `  1 )  ->  A  =/=  C
) )
9273, 91syld 42 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  0 ) )  =  ( E `
 ( F ` 
1 ) )  ->  A  =/=  C ) )
9354, 92syl5 30 . . . . . . . . . . . 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 ) )
9493adantl 453 . . . . . . . . . . 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 ) )
9553, 94sylbid 207 . . . . . . . . . 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 ) )
9695adantrd 455 . . . . . . . . 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 ) )
9796expimpd 587 . . . . . . . 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
) )
98 ax-1 5 . . . . . . . 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
) )
9997, 98pm2.61ine 2627 . . . . . . 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
)
10099adantl 453 . . . . . 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 )
101 usgraedgrn 21268 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  D  =/=  A )
102101necomd 2634 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { D ,  A }  e.  ran  E )  ->  A  =/=  D )
103102ex 424 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { D ,  A }  e.  ran  E  ->  A  =/=  D
) )
104103ad2antrr 707 . . . . . . . . . 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
) )
105104com12 29 . . . . . . . . 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
) )
106105adantl 453 . . . . . . . 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
) )
107106adantl 453 . . . . . . 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
) )
108107imp 419 . . . . . 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 )
10946, 100, 1083jca 1134 . . . . 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 ) )
110 usgraedgrn 21268 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  B  =/=  C )
111110ex 424 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  B  =/=  C
) )
112111ad2antrr 707 . . . . . . . . . 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
) )
113112com12 29 . . . . . . . . 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
) )
114113adantl 453 . . . . . . . 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
) )
115114adantr 452 . . . . . . 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
) )
116115imp 419 . . . . . 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 )
117 prcom 3826 . . . . . . . . . . . . . . . . . . . . 21  |-  { D ,  A }  =  { A ,  D }
118117eqeq2i 2398 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( F `
 3 ) )  =  { D ,  A }  <->  ( E `  ( F `  3 ) )  =  { A ,  D } )
119118a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  D  ->  (
( E `  ( F `  3 )
)  =  { D ,  A }  <->  ( E `  ( F `  3
) )  =  { A ,  D }
) )
120 preq2 3828 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  D  ->  { A ,  B }  =  { A ,  D }
)
121120eqeq2d 2399 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  D  ->  (
( E `  ( F `  0 )
)  =  { A ,  B }  <->  ( E `  ( F `  0
) )  =  { A ,  D }
) )
122119, 121anbi12d 692 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  D  ->  (
( ( E `  ( F `  3 ) )  =  { D ,  A }  /\  ( E `  ( F `  0 ) )  =  { A ,  B } )  <->  ( ( E `  ( F `  3 ) )  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } ) ) )
123122adantr 452 . . . . . . . . . . . . . . . . 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 } ) ) )
124 eqtr3 2407 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E `  ( F `  3 )
)  =  { A ,  D }  /\  ( E `  ( F `  0 ) )  =  { A ,  D } )  ->  ( E `  ( F `  3 ) )  =  ( E `  ( F `  0 ) ) )
125 elfzo0 11102 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 3  e.  ( 0..^ 4 )  <->  ( 3  e. 
NN0  /\  4  e.  NN  /\  3  <  4
) )
12627, 59, 24, 125mpbir3an 1136 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  3  e.  ( 0..^ 4 )
127 ffvelrn 5808 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F : ( 0..^ 4 ) --> dom  E  /\  3  e.  (
0..^ 4 ) )  ->  ( F ` 
3 )  e.  dom  E )
128126, 127mpan2 653 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( F `  3 )  e.  dom  E )
129128, 63jca 519 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( F : ( 0..^ 4 ) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) )
13058, 129syl6bi 220 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( F `  3
)  e.  dom  E  /\  ( F `  0
)  e.  dom  E
) ) )
13156, 130mpan9 456 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  4 )  ->  ( ( F `
 3 )  e. 
dom  E  /\  ( F `  0 )  e.  dom  E ) )
1321313adant2 976 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  e.  dom  E  /\  ( F ` 
0 )  e.  dom  E ) )
133 f1veqaeq 5945 . . . . . . . . . . . . . . . . . . . . 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
) ) )
13455, 132, 133syl2an 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  3 ) )  =  ( E `
 ( F ` 
0 ) )  -> 
( F `  3
)  =  ( F `
 0 ) ) )
135 f1veqaeq 5945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F : ( 0..^ 4 ) -1-1-> dom  E  /\  ( 3  e.  ( 0..^ 4 )  /\  0  e.  ( 0..^ 4 ) ) )  ->  ( ( F `
 3 )  =  ( F `  0
)  ->  3  = 
0 ) )
136126, 61, 135mpanr12 667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  -> 
3  =  0 ) )
137 3ne0 10018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  3  =/=  0
138 df-ne 2553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( 3  =/=  0  <->  -.  3  =  0 )
139137, 138mpbi 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  -.  3  =  0
140139pm2.21i 125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 3  =  0  ->  B  =/=  D )
141136, 140syl6 31 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( F : ( 0..^ 4 ) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) )
14276, 141syl6bi 220 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  4  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) ) )
143142com12 29 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  (
( # `  F )  =  4  ->  (
( F `  3
)  =  ( F `
 0 )  ->  B  =/=  D ) ) )
14474, 143sylbir 205 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  Fun  `' F )  ->  ( ( # `  F )  =  4  ->  ( ( F `
 3 )  =  ( F `  0
)  ->  B  =/=  D ) ) )
145144ex 424 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) ) ) )
14656, 145syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( F  e. Word  dom  E  ->  ( Fun  `' F  -> 
( ( # `  F
)  =  4  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) ) ) )
1471463imp 1147 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 )  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) )
148147adantl 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( F ` 
3 )  =  ( F `  0 )  ->  B  =/=  D
) )
149134, 148syld 42 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( E `  ( F `  3 ) )  =  ( E `
 ( F ` 
0 ) )  ->  B  =/=  D ) )
150124, 149syl5 30 . . . . . . . . . . . . . . . . . 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 ) )
151150adantl 453 . . . . . . . . . . . . . . . . 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 ) )
152123, 151sylbid 207 . . . . . . . . . . . . . . . 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 ) )
153152com12 29 . . . . . . . . . . . . . . 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 ) )
154153ex 424 . . . . . . . . . . . . . 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
) ) )
155154adantl 453 . . . . . . . . . . . . 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 ) ) )
156155com12 29 . . . . . . . . . . . 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 ) ) )
157156adantr 452 . . . . . . . . . . 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 ) ) )
158157imp 419 . . . . . . . . . 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 ) )
159158com12 29 . . . . . . . . 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 ) )
160159expimpd 587 . . . . . . . 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
) )
161 ax-1 5 . . . . . . . 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
) )
162160, 161pm2.61ine 2627 . . . . . . 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
)
163162adantl 453 . . . . . 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 )
164 usgraedgrn 21268 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { C ,  D }  e.  ran  E )  ->  C  =/=  D )
165164ex 424 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( { C ,  D }  e.  ran  E  ->  C  =/=  D
) )
166165ad2antrr 707 . . . . . . . . . 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
) )
167166com12 29 . . . . . . . . 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
) )
168167adantr 452 . . . . . . . 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
) )
169168adantl 453 . . . . . . 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
) )
170169imp 419 . . . . . 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 )
171116, 163, 1703jca 1134 . . . . 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 ) )
172109, 171jca 519 . . . 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 ) ) )
17340, 172jca 519 . . 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 ) ) ) )
17439, 173mpancom 651 . 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 ) ) ) )
175174ex 424 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   {cpr 3759   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   ran crn 4820   Fun wfun 5389   -->wf 5391   -1-1->wf1 5392   ` cfv 5395  (class class class)co 6021   0cc0 8924   1c1 8925    < clt 9054   NNcn 9933   2c2 9982   3c3 9983   4c4 9984   NN0cn0 10154  ..^cfzo 11066   #chash 11546  Word cword 11645   USGrph cusg 21233
This theorem is referenced by:  4cycl4dv4e  21504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-hash 11547  df-word 11651  df-usgra 21235
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