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Theorem nb3graprlem1 25258
Description: Lemma 1 for nb3grapr 25260. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
Assertion
Ref Expression
nb3graprlem1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )

Proof of Theorem nb3graprlem1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 prid1g 4069 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
213ad2ant2 1052 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
32adantr 472 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { B ,  C } )
43adantr 472 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  B  e.  { B ,  C }
)
5 eleq2 2538 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
65eqcoms 2479 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
76adantl 473 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( B  e.  { B ,  C } 
<->  B  e.  ( <. V ,  E >. Neighbors  A
) ) )
84, 7mpbid 215 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  B  e.  ( <. V ,  E >. Neighbors  A ) )
9 nbgraeledg 25237 . . . . . . 7  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { B ,  A }  e.  ran  E ) )
10 prcom 4041 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1110a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { B ,  A }  =  { A ,  B }
)
1211eleq1d 2533 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  A }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
139, 12bitrd 261 . . . . . 6  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
1413adantl 473 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( B  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  B }  e.  ran  E ) )
1514ad2antlr 741 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
168, 15mpbid 215 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  { A ,  B }  e.  ran  E )
17 prid2g 4070 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
18173ad2ant3 1053 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
1918adantr 472 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { B ,  C } )
2019adantr 472 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  C  e.  { B ,  C }
)
21 eleq2 2538 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2221eqcoms 2479 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2322adantl 473 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( C  e.  { B ,  C } 
<->  C  e.  ( <. V ,  E >. Neighbors  A
) ) )
2420, 23mpbid 215 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  C  e.  ( <. V ,  E >. Neighbors  A ) )
25 nbgraeledg 25237 . . . . . . 7  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { C ,  A }  e.  ran  E ) )
26 prcom 4041 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
2726a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { C ,  A }  =  { A ,  C }
)
2827eleq1d 2533 . . . . . . 7  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E ) )
2925, 28bitrd 261 . . . . . 6  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3029adantl 473 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( C  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  C }  e.  ran  E ) )
3130ad2antlr 741 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3224, 31mpbid 215 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  { A ,  C }  e.  ran  E )
3316, 32jca 541 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
34 nbusgra 25235 . . . . 5  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  A )  =  {
v  e.  V  |  { A ,  v }  e.  ran  E }
)
3534ad2antll 743 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { v  e.  V  |  { A ,  v }  e.  ran  E } )
3635adantr 472 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { v  e.  V  |  { A ,  v }  e.  ran  E } )
37 eleq2 2538 . . . . . . . . . 10  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
3837ad2antrl 742 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
3938adantr 472 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
40 vex 3034 . . . . . . . . . . 11  |-  v  e. 
_V
4140eltp 4008 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
42 usgraedgrn 25187 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  A  =/=  v )
43 df-ne 2643 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
44 pm2.24 112 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4544eqcoms 2479 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  A  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4645com12 31 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  =  v  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) )
4743, 46sylbi 200 . . . . . . . . . . . . . . . 16  |-  ( A  =/=  v  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4842, 47syl 17 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4948ex 441 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5049ad2antll 743 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5150adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5251com3r 81 . . . . . . . . . . 11  |-  ( v  =  A  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
53 orc 392 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  (
v  =  B  \/  v  =  C )
)
5453a1d 25 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) )
5554a1d 25 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
56 olc 391 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  (
v  =  B  \/  v  =  C )
)
5756a1d 25 . . . . . . . . . . . 12  |-  ( v  =  C  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) )
5857a1d 25 . . . . . . . . . . 11  |-  ( v  =  C  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
5952, 55, 583jaoi 1357 . . . . . . . . . 10  |-  ( ( v  =  A  \/  v  =  B  \/  v  =  C )  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6041, 59sylbi 200 . . . . . . . . 9  |-  ( v  e.  { A ,  B ,  C }  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6160com12 31 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  { A ,  B ,  C }  ->  ( { A , 
v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6239, 61sylbid 223 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  -> 
( { A , 
v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6362impd 438 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  ran  E
)  ->  ( v  =  B  \/  v  =  C ) ) )
64 eqid 2471 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
65643mix2i 1203 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
66 eltpg 4005 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
6765, 66mpbiri 241 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  Y  ->  B  e.  { A ,  B ,  C } )
6867a1d 25 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  Y  ->  (
( V  =  { A ,  B ,  C }  /\  V USGrph  E
)  ->  B  e.  { A ,  B ,  C } ) )
69683ad2ant2 1052 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  B  e.  { A ,  B ,  C } ) )
7069imp 436 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { A ,  B ,  C } )
7170adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  B  e.  { A ,  B ,  C } )
72 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7372bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7473adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C } ) )
7571, 74mpbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  { A ,  B ,  C } )
7637bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
7776ad2antrl 742 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
7877adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
7975, 78mpbid 215 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  V )
8079ex 441 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8180adantr 472 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8281impcom 437 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
v  e.  V )
83 preq2 4043 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8483eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8584eqcoms 2479 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8685biimpcd 232 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  ran  E  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8786ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8887impcom 437 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  ->  { A ,  v }  e.  ran  E )
8982, 88jca 541 . . . . . . . . 9  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  ran  E
) )
9089ex 441 . . . . . . . 8  |-  ( v  =  B  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
91 tpid3g 4078 . . . . . . . . . . . . . . . . . . 19  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
9291a1d 25 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  Z  ->  (
( V  =  { A ,  B ,  C }  /\  V USGrph  E
)  ->  C  e.  { A ,  B ,  C } ) )
93923ad2ant3 1053 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  C  e.  { A ,  B ,  C } ) )
9493imp 436 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { A ,  B ,  C } )
9594adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  C  e.  { A ,  B ,  C } )
96 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9796bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9897adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C } ) )
9995, 98mpbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  { A ,  B ,  C } )
10077adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
10199, 100mpbid 215 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  V )
102101ex 441 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
103102adantr 472 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
104103impcom 437 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
v  e.  V )
105 preq2 4043 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
106105eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
107106eqcoms 2479 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
108107biimpcd 232 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  ran  E  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
109108ad2antll 743 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
110109impcom 437 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  ->  { A ,  v }  e.  ran  E )
111104, 110jca 541 . . . . . . . . 9  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  ran  E
) )
112111ex 441 . . . . . . . 8  |-  ( v  =  C  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
11390, 112jaoi 386 . . . . . . 7  |-  ( ( v  =  B  \/  v  =  C )  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
114113com12 31 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  =  B  \/  v  =  C )  ->  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
11563, 114impbid 195 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  ran  E
)  <->  ( v  =  B  \/  v  =  C ) ) )
116115abbidv 2589 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }  =  { v  |  ( v  =  B  \/  v  =  C ) } )
117 df-rab 2765 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }
118 dfpr2 3974 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
119116, 117, 1183eqtr4g 2530 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { B ,  C }
)
12036, 119eqtrd 2505 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { B ,  C } )
12133, 120impbida 850 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   {crab 2760   {cpr 3961   {ctp 3963   <.cop 3965   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227
This theorem is referenced by:  nb3grapr  25260  nb3grapr2  25261  nb3gra2nb  25262
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