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Theorem usgr2edg 39291
Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Hypotheses
Ref Expression
usgrf1oedg.i  |-  I  =  (iEdg `  G )
usgrf1oedg.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
usgr2edg  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) )
Distinct variable groups:    x, G    x, A, y    x, B, y    y, G    x, I, y    x, N, y
Allowed substitution hints:    E( x, y)

Proof of Theorem usgr2edg
StepHypRef Expression
1 usgrf1oedg.e . . . . 5  |-  E  =  (Edg `  G )
2 eqid 2451 . . . . 5  |-  (Vtx `  G )  =  (Vtx
`  G )
31, 2usgrpredgav 39280 . . . 4  |-  ( ( G  e. USGraph  /\  { N ,  A }  e.  E
)  ->  ( N  e.  (Vtx `  G )  /\  A  e.  (Vtx `  G ) ) )
43ad2ant2r 753 . . 3  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) ) )
51, 2usgrpredgav 39280 . . . 4  |-  ( ( G  e. USGraph  /\  { B ,  N }  e.  E
)  ->  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) )
65ad2ant2rl 755 . . 3  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )
7 pm3.2 449 . . . 4  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  (
( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )  ->  ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) ) ) )
87impancom 442 . . 3  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  (
( ( N  e.  (Vtx `  G )  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G )
) )  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) ) ) )
94, 6, 8mp2and 685 . 2  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) ) )
101a1i 11 . . . . . . . . 9  |-  ( G  e. USGraph  ->  E  =  (Edg
`  G ) )
11 edgaval 39212 . . . . . . . . 9  |-  ( G  e. USGraph  ->  (Edg `  G
)  =  ran  (iEdg `  G ) )
12 usgrf1oedg.i . . . . . . . . . . . 12  |-  I  =  (iEdg `  G )
1312eqcomi 2460 . . . . . . . . . . 11  |-  (iEdg `  G )  =  I
1413a1i 11 . . . . . . . . . 10  |-  ( G  e. USGraph  ->  (iEdg `  G
)  =  I )
1514rneqd 5062 . . . . . . . . 9  |-  ( G  e. USGraph  ->  ran  (iEdg `  G
)  =  ran  I
)
1610, 11, 153eqtrd 2489 . . . . . . . 8  |-  ( G  e. USGraph  ->  E  =  ran  I )
1716eleq2d 2514 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { N ,  A }  e.  E  <->  { N ,  A }  e.  ran  I ) )
1816eleq2d 2514 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { B ,  N }  e.  E  <->  { B ,  N }  e.  ran  I ) )
1917, 18anbi12d 717 . . . . . 6  |-  ( G  e. USGraph  ->  ( ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
)  <->  ( { N ,  A }  e.  ran  I  /\  { B ,  N }  e.  ran  I ) ) )
20 usgrfun 39245 . . . . . . . . 9  |-  ( G  e. USGraph  ->  Fun  (iEdg `  G
) )
2112funeqi 5602 . . . . . . . . 9  |-  ( Fun  I  <->  Fun  (iEdg `  G
) )
2220, 21sylibr 216 . . . . . . . 8  |-  ( G  e. USGraph  ->  Fun  I )
23 funfn 5611 . . . . . . . 8  |-  ( Fun  I  <->  I  Fn  dom  I )
2422, 23sylib 200 . . . . . . 7  |-  ( G  e. USGraph  ->  I  Fn  dom  I )
25 fvelrnb 5912 . . . . . . . 8  |-  ( I  Fn  dom  I  -> 
( { N ,  A }  e.  ran  I 
<->  E. x  e.  dom  I ( I `  x )  =  { N ,  A }
) )
26 fvelrnb 5912 . . . . . . . 8  |-  ( I  Fn  dom  I  -> 
( { B ,  N }  e.  ran  I 
<->  E. y  e.  dom  I ( I `  y )  =  { B ,  N }
) )
2725, 26anbi12d 717 . . . . . . 7  |-  ( I  Fn  dom  I  -> 
( ( { N ,  A }  e.  ran  I  /\  { B ,  N }  e.  ran  I )  <->  ( E. x  e.  dom  I ( I `  x )  =  { N ,  A }  /\  E. y  e.  dom  I ( I `
 y )  =  { B ,  N } ) ) )
2824, 27syl 17 . . . . . 6  |-  ( G  e. USGraph  ->  ( ( { N ,  A }  e.  ran  I  /\  { B ,  N }  e.  ran  I )  <->  ( E. x  e.  dom  I ( I `  x )  =  { N ,  A }  /\  E. y  e.  dom  I ( I `
 y )  =  { B ,  N } ) ) )
2919, 28bitrd 257 . . . . 5  |-  ( G  e. USGraph  ->  ( ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
)  <->  ( E. x  e.  dom  I ( I `
 x )  =  { N ,  A }  /\  E. y  e. 
dom  I ( I `
 y )  =  { B ,  N } ) ) )
3029ad2antrr 732 . . . 4  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  (
( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )  <->  ( E. x  e.  dom  I ( I `  x )  =  { N ,  A }  /\  E. y  e.  dom  I ( I `  y )  =  { B ,  N }
) ) )
31 reeanv 2958 . . . . 5  |-  ( E. x  e.  dom  I E. y  e.  dom  I ( ( I `
 x )  =  { N ,  A }  /\  ( I `  y )  =  { B ,  N }
)  <->  ( E. x  e.  dom  I ( I `
 x )  =  { N ,  A }  /\  E. y  e. 
dom  I ( I `
 y )  =  { B ,  N } ) )
32 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
I `  x )  =  ( I `  y ) )
3332eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( I `  x
)  =  { N ,  A }  <->  ( I `  y )  =  { N ,  A }
) )
3433anbi1d 711 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( I `  x )  =  { N ,  A }  /\  ( I `  y
)  =  { B ,  N } )  <->  ( (
I `  y )  =  { N ,  A }  /\  ( I `  y )  =  { B ,  N }
) ) )
35 eqtr2 2471 . . . . . . . . . . . . . 14  |-  ( ( ( I `  y
)  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } )  ->  { N ,  A }  =  { B ,  N }
)
36 prcom 4050 . . . . . . . . . . . . . . . 16  |-  { B ,  N }  =  { N ,  B }
3736eqeq2i 2463 . . . . . . . . . . . . . . 15  |-  ( { N ,  A }  =  { B ,  N } 
<->  { N ,  A }  =  { N ,  B } )
38 preq12bg 4154 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( N  e.  (Vtx `  G
)  /\  B  e.  (Vtx `  G ) ) )  ->  ( { N ,  A }  =  { N ,  B } 
<->  ( ( N  =  N  /\  A  =  B )  \/  ( N  =  B  /\  A  =  N )
) ) )
3938ancom2s 811 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )  ->  ( { N ,  A }  =  { N ,  B } 
<->  ( ( N  =  N  /\  A  =  B )  \/  ( N  =  B  /\  A  =  N )
) ) )
40 eqneqall 2634 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  =  B  ->  ( A  =/=  B  ->  x  =/=  y ) )
4140adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  =  N  /\  A  =  B )  ->  ( A  =/=  B  ->  x  =/=  y ) )
42 eqtr 2470 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  =  N  /\  N  =  B )  ->  A  =  B )
4342ancoms 455 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  =  B  /\  A  =  N )  ->  A  =  B )
4443, 40syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  =  B  /\  A  =  N )  ->  ( A  =/=  B  ->  x  =/=  y ) )
4541, 44jaoi 381 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  =  N  /\  A  =  B )  \/  ( N  =  B  /\  A  =  N ) )  -> 
( A  =/=  B  ->  x  =/=  y ) )
4645adantld 469 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  =  N  /\  A  =  B )  \/  ( N  =  B  /\  A  =  N ) )  -> 
( ( G  e. USGraph  /\  A  =/=  B
)  ->  x  =/=  y ) )
4739, 46syl6bi 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )  ->  ( { N ,  A }  =  { N ,  B }  ->  ( ( G  e. USGraph  /\  A  =/=  B
)  ->  x  =/=  y ) ) )
4847com3l 84 . . . . . . . . . . . . . . . 16  |-  ( { N ,  A }  =  { N ,  B }  ->  ( ( G  e. USGraph  /\  A  =/=  B
)  ->  ( (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )  ->  x  =/=  y ) ) )
4948impd 433 . . . . . . . . . . . . . . 15  |-  ( { N ,  A }  =  { N ,  B }  ->  ( ( ( G  e. USGraph  /\  A  =/= 
B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  x  =/=  y ) )
5037, 49sylbi 199 . . . . . . . . . . . . . 14  |-  ( { N ,  A }  =  { B ,  N }  ->  ( ( ( G  e. USGraph  /\  A  =/= 
B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  x  =/=  y ) )
5135, 50syl 17 . . . . . . . . . . . . 13  |-  ( ( ( I `  y
)  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } )  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  ->  x  =/=  y ) )
5234, 51syl6bi 232 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( ( I `  x )  =  { N ,  A }  /\  ( I `  y
)  =  { B ,  N } )  -> 
( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  ->  x  =/=  y ) ) )
5352com23 81 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  ->  ( (
( I `  x
)  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } )  ->  x  =/=  y ) ) )
5453impd 433 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  ->  x  =/=  y ) )
55 ax-1 6 . . . . . . . . . 10  |-  ( x  =/=  y  ->  (
( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  ->  x  =/=  y ) )
5654, 55pm2.61ine 2707 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  ->  x  =/=  y )
57 prid1g 4078 . . . . . . . . . . . . . 14  |-  ( N  e.  (Vtx `  G
)  ->  N  e.  { N ,  A }
)
5857ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )  ->  N  e.  { N ,  A }
)
5958adantl 468 . . . . . . . . . . . 12  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  { N ,  A } )
60 eleq2 2518 . . . . . . . . . . . 12  |-  ( ( I `  x )  =  { N ,  A }  ->  ( N  e.  ( I `  x )  <->  N  e.  { N ,  A }
) )
6159, 60syl5ibr 225 . . . . . . . . . . 11  |-  ( ( I `  x )  =  { N ,  A }  ->  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  ( I `  x
) ) )
6261adantr 467 . . . . . . . . . 10  |-  ( ( ( I `  x
)  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } )  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  ( I `  x
) ) )
6362impcom 432 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  ->  N  e.  ( I `  x ) )
64 prid2g 4079 . . . . . . . . . . . . . 14  |-  ( N  e.  (Vtx `  G
)  ->  N  e.  { B ,  N }
)
6564ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) )  ->  N  e.  { B ,  N }
)
6665adantl 468 . . . . . . . . . . . 12  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  { B ,  N } )
67 eleq2 2518 . . . . . . . . . . . 12  |-  ( ( I `  y )  =  { B ,  N }  ->  ( N  e.  ( I `  y )  <->  N  e.  { B ,  N }
) )
6866, 67syl5ibr 225 . . . . . . . . . . 11  |-  ( ( I `  y )  =  { B ,  N }  ->  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  ( I `  y
) ) )
6968adantl 468 . . . . . . . . . 10  |-  ( ( ( I `  x
)  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } )  ->  (
( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  ->  N  e.  ( I `  y
) ) )
7069impcom 432 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  ->  N  e.  ( I `  y ) )
7156, 63, 703jca 1188 . . . . . . . 8  |-  ( ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( ( I `  x )  =  { N ,  A }  /\  (
I `  y )  =  { B ,  N } ) )  -> 
( x  =/=  y  /\  N  e.  (
I `  x )  /\  N  e.  (
I `  y )
) )
7271ex 436 . . . . . . 7  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  (
( ( I `  x )  =  { N ,  A }  /\  ( I `  y
)  =  { B ,  N } )  -> 
( x  =/=  y  /\  N  e.  (
I `  x )  /\  N  e.  (
I `  y )
) ) )
7372reximdv 2861 . . . . . 6  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  ( E. y  e.  dom  I ( ( I `
 x )  =  { N ,  A }  /\  ( I `  y )  =  { B ,  N }
)  ->  E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) ) )
7473reximdv 2861 . . . . 5  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  ( E. x  e.  dom  I E. y  e.  dom  I ( ( I `
 x )  =  { N ,  A }  /\  ( I `  y )  =  { B ,  N }
)  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) ) )
7531, 74syl5bir 222 . . . 4  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  (
( E. x  e. 
dom  I ( I `
 x )  =  { N ,  A }  /\  E. y  e. 
dom  I ( I `
 y )  =  { B ,  N } )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) ) )
7630, 75sylbid 219 . . 3  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  (
( N  e.  (Vtx
`  G )  /\  A  e.  (Vtx `  G
) )  /\  ( B  e.  (Vtx `  G
)  /\  N  e.  (Vtx `  G ) ) ) )  ->  (
( { N ,  A }  e.  E  /\  { B ,  N }  e.  E )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) ) )
7776imp 431 . 2  |-  ( ( ( ( G  e. USGraph  /\  A  =/=  B
)  /\  ( ( N  e.  (Vtx `  G
)  /\  A  e.  (Vtx `  G ) )  /\  ( B  e.  (Vtx `  G )  /\  N  e.  (Vtx `  G ) ) ) )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) )
789, 77syl 17 1  |-  ( ( ( G  e. USGraph  /\  A  =/=  B )  /\  ( { N ,  A }  e.  E  /\  { B ,  N }  e.  E
) )  ->  E. x  e.  dom  I E. y  e.  dom  I ( x  =/=  y  /\  N  e.  ( I `  x
)  /\  N  e.  ( I `  y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   {cpr 3970   dom cdm 4834   ran crn 4835   Fun wfun 5576    Fn wfn 5577   ` cfv 5582  Vtxcvtx 39101  iEdgciedg 39102  Edgcedga 39210   USGraph cusgr 39236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  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 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  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-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-umgr 39175  df-edga 39211  df-usgr 39238
This theorem is referenced by:  usgr2edg1  39292
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