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Theorem nb3graprlem1 23371
Description: Lemma 1 for nb3grapr 23373. (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 3993 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
213ad2ant2 1010 . . . . . . 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 2504 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
65eqcoms 2446 . . . . . 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 23353 . . . . . . 7  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { B ,  A }  e.  ran  E ) )
10 prcom 3965 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1110a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { B ,  A }  =  { A ,  B }
)
1211eleq1d 2509 . . . . . . 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 3994 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
18173ad2ant3 1011 . . . . . . 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 2504 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2221eqcoms 2446 . . . . . 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 23353 . . . . . . 7  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { C ,  A }  e.  ran  E ) )
26 prcom 3965 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
2726a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { C ,  A }  =  { A ,  C }
)
2827eleq1d 2509 . . . . . . 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 23351 . . . . 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 2504 . . . . . . . . . 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 2987 . . . . . . . . . . 11  |-  v  e. 
_V
4140eltp 3933 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
42 usgraedgrn 23312 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  A  =/=  v )
43 df-ne 2620 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
44 pm2.24 109 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4544eqcoms 2446 . . . . . . . . . . . . . . . . . 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 1281 . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
65643mix2i 1161 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
66 eltpg 3930 . . . . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . 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 2503 . . . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8483eleq1d 2509 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8584eqcoms 2446 . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . 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 2503 . . . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
106105eleq1d 2509 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
107106eqcoms 2446 . . . . . . . . . . . . 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 2563 . . . 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 2736 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }
118 dfpr2 3904 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
119116, 117, 1183eqtr4g 2500 . . 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 2475 . 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 828 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 964    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2618   {crab 2731   {cpr 3891   {ctp 3893   <.cop 3895   class class class wbr 4304   ran crn 4853  (class class class)co 6103   USGrph cusg 23276   Neighbors cnbgra 23341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-hash 12116  df-usgra 23278  df-nbgra 23344
This theorem is referenced by:  nb3grapr  23373  nb3grapr2  23374  nb3gra2nb  23375
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