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Theorem nb3graprlem1 24283
Description: Lemma 1 for nb3grapr 24285. (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 4139 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
213ad2ant2 1018 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
32adantr 465 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { B ,  C } )
43adantr 465 . . . . 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 2540 . . . . . . 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 466 . . . . 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 210 . . . 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 24262 . . . . . . 7  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { B ,  A }  e.  ran  E ) )
10 prcom 4111 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1110a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { B ,  A }  =  { A ,  B }
)
1211eleq1d 2536 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  A }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
139, 12bitrd 253 . . . . . 6  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
1413adantl 466 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( B  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  B }  e.  ran  E ) )
1514ad2antlr 726 . . . 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 210 . . 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 4140 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
18173ad2ant3 1019 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
1918adantr 465 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { B ,  C } )
2019adantr 465 . . . . 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 2540 . . . . . . 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 466 . . . . 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 210 . . . 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 24262 . . . . . . 7  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { C ,  A }  e.  ran  E ) )
26 prcom 4111 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
2726a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { C ,  A }  =  { A ,  C }
)
2827eleq1d 2536 . . . . . . 7  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E ) )
2925, 28bitrd 253 . . . . . 6  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3029adantl 466 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( C  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  C }  e.  ran  E ) )
3130ad2antlr 726 . . . 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 210 . . 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 532 . 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 24260 . . . . 5  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  A )  =  {
v  e.  V  |  { A ,  v }  e.  ran  E }
)
3534ad2antll 728 . . . 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 465 . . 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 2540 . . . . . . . . . 10  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
3837ad2antrl 727 . . . . . . . . 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 465 . . . . . . . 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 3121 . . . . . . . . . . 11  |-  v  e. 
_V
4140eltp 4078 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
42 usgraedgrn 24213 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  A  =/=  v )
43 df-ne 2664 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
44 pm2.24 109 . . . . . . . . . . . . . . . . . . 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 195 . . . . . . . . . . . . . . . 16  |-  ( A  =/=  v  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4842, 47syl 16 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4948ex 434 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5049ad2antll 728 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 79 . . . . . . . . . . 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 385 . . . . . . . . . . . . 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 384 . . . . . . . . . . . . 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 1291 . . . . . . . . . 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 195 . . . . . . . . 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 215 . . . . . . 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 431 . . . . . 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 2467 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
65643mix2i 1169 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
66 eltpg 4075 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
6765, 66mpbiri 233 . . . . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  B  e.  { A ,  B ,  C } ) )
7069imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { A ,  B ,  C } )
7170adantr 465 . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7372bicomd 201 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7473adantl 466 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . . 16  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
7776ad2antrl 727 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  V )
8079ex 434 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8180adantr 465 . . . . . . . . . . 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 430 . . . . . . . . . 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 4113 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8483eleq1d 2536 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  ran  E  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8786ad2antrl 727 . . . . . . . . . . 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 430 . . . . . . . . . 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 532 . . . . . . . . 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 434 . . . . . . . 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 4148 . . . . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  C  e.  { A ,  B ,  C } ) )
9493imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { A ,  B ,  C } )
9594adantr 465 . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9796bicomd 201 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9897adantl 466 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  V )
102101ex 434 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
103102adantr 465 . . . . . . . . . . 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 430 . . . . . . . . . 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 4113 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
106105eleq1d 2536 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  ran  E  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
109108ad2antll 728 . . . . . . . . . . 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 430 . . . . . . . . . 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 532 . . . . . . . . 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 434 . . . . . . . 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 379 . . . . . . 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 191 . . . . 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 2603 . . . 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 2826 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }
118 dfpr2 4048 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
119116, 117, 1183eqtr4g 2533 . . 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 2508 . 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 830 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 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   {crab 2821   {cpr 4035   {ctp 4037   <.cop 4039   class class class wbr 4453   ran crn 5006  (class class class)co 6295   USGrph cusg 24162   Neighbors cnbgra 24249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-usgra 24165  df-nbgra 24252
This theorem is referenced by:  nb3grapr  24285  nb3grapr2  24286  nb3gra2nb  24287
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