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Theorem nb3grprlem1 39454
Description: Lemma 1 for nb3grapr 25181. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v  |-  V  =  (Vtx `  G )
nb3grpr.e  |-  E  =  (Edg `  G )
nb3grpr.g  |-  ( ph  ->  G  e. USGraph  )
nb3grpr.t  |-  ( ph  ->  V  =  { A ,  B ,  C }
)
nb3grpr.s  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
Assertion
Ref Expression
nb3grprlem1  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E )
) )

Proof of Theorem nb3grprlem1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nb3grpr.s . . . . . . 7  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
2 prid1g 4078 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
323ad2ant2 1030 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
41, 3syl 17 . . . . . 6  |-  ( ph  ->  B  e.  { B ,  C } )
54adantr 467 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  B  e.  { B ,  C } )
6 eleq2 2518 . . . . . . 7  |-  ( { B ,  C }  =  ( G NeighbVtx  A )  ->  ( B  e. 
{ B ,  C } 
<->  B  e.  ( G NeighbVtx  A ) ) )
76eqcoms 2459 . . . . . 6  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  ->  ( B  e.  { B ,  C }  <->  B  e.  ( G NeighbVtx  A ) ) )
87adantl 468 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( B  e.  { B ,  C }  <->  B  e.  ( G NeighbVtx  A ) ) )
95, 8mpbid 214 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  B  e.  ( G NeighbVtx  A ) )
10 nb3grpr.g . . . . . 6  |-  ( ph  ->  G  e. USGraph  )
11 nb3grpr.e . . . . . . . 8  |-  E  =  (Edg `  G )
1211nbusgreledg 39421 . . . . . . 7  |-  ( G  e. USGraph  ->  ( B  e.  ( G NeighbVtx  A )  <->  { B ,  A }  e.  E ) )
13 prcom 4050 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1413a1i 11 . . . . . . . 8  |-  ( G  e. USGraph  ->  { B ,  A }  =  { A ,  B }
)
1514eleq1d 2513 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { B ,  A }  e.  E  <->  { A ,  B }  e.  E ) )
1612, 15bitrd 257 . . . . . 6  |-  ( G  e. USGraph  ->  ( B  e.  ( G NeighbVtx  A )  <->  { A ,  B }  e.  E ) )
1710, 16syl 17 . . . . 5  |-  ( ph  ->  ( B  e.  ( G NeighbVtx  A )  <->  { A ,  B }  e.  E
) )
1817adantr 467 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( B  e.  ( G NeighbVtx  A )  <->  { A ,  B }  e.  E
) )
199, 18mpbid 214 . . 3  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  { A ,  B }  e.  E )
20 prid2g 4079 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
21203ad2ant3 1031 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
221, 21syl 17 . . . . . 6  |-  ( ph  ->  C  e.  { B ,  C } )
2322adantr 467 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  C  e.  { B ,  C } )
24 eleq2 2518 . . . . . . 7  |-  ( { B ,  C }  =  ( G NeighbVtx  A )  ->  ( C  e. 
{ B ,  C } 
<->  C  e.  ( G NeighbVtx  A ) ) )
2524eqcoms 2459 . . . . . 6  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  ->  ( C  e.  { B ,  C }  <->  C  e.  ( G NeighbVtx  A ) ) )
2625adantl 468 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( C  e.  { B ,  C }  <->  C  e.  ( G NeighbVtx  A ) ) )
2723, 26mpbid 214 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  C  e.  ( G NeighbVtx  A ) )
2811nbusgreledg 39421 . . . . . . 7  |-  ( G  e. USGraph  ->  ( C  e.  ( G NeighbVtx  A )  <->  { C ,  A }  e.  E ) )
29 prcom 4050 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
3029a1i 11 . . . . . . . 8  |-  ( G  e. USGraph  ->  { C ,  A }  =  { A ,  C }
)
3130eleq1d 2513 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { C ,  A }  e.  E  <->  { A ,  C }  e.  E ) )
3228, 31bitrd 257 . . . . . 6  |-  ( G  e. USGraph  ->  ( C  e.  ( G NeighbVtx  A )  <->  { A ,  C }  e.  E ) )
3310, 32syl 17 . . . . 5  |-  ( ph  ->  ( C  e.  ( G NeighbVtx  A )  <->  { A ,  C }  e.  E
) )
3433adantr 467 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( C  e.  ( G NeighbVtx  A )  <->  { A ,  C }  e.  E
) )
3527, 34mpbid 214 . . 3  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  { A ,  C }  e.  E )
3619, 35jca 535 . 2  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( { A ,  B }  e.  E  /\  { A ,  C }  e.  E )
)
37 nb3grpr.v . . . . . 6  |-  V  =  (Vtx `  G )
3837, 11nbusgr 39417 . . . . 5  |-  ( G  e. USGraph  ->  ( G NeighbVtx  A )  =  { v  e.  V  |  { A ,  v }  e.  E } )
3910, 38syl 17 . . . 4  |-  ( ph  ->  ( G NeighbVtx  A )  =  { v  e.  V  |  { A ,  v }  e.  E }
)
4039adantr 467 . . 3  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( G NeighbVtx  A )  =  {
v  e.  V  |  { A ,  v }  e.  E } )
41 nb3grpr.t . . . . . . . . . 10  |-  ( ph  ->  V  =  { A ,  B ,  C }
)
42 eleq2 2518 . . . . . . . . . 10  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
4341, 42syl 17 . . . . . . . . 9  |-  ( ph  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
4443adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
45 vex 3048 . . . . . . . . . . 11  |-  v  e. 
_V
4645eltp 4017 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
4711usgredgne 39289 . . . . . . . . . . . . . . . 16  |-  ( ( G  e. USGraph  /\  { A ,  v }  e.  E )  ->  A  =/=  v )
48 df-ne 2624 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
49 pm2.24 113 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
5049eqcoms 2459 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  A  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
5150com12 32 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  =  v  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) )
5248, 51sylbi 199 . . . . . . . . . . . . . . . 16  |-  ( A  =/=  v  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
5347, 52syl 17 . . . . . . . . . . . . . . 15  |-  ( ( G  e. USGraph  /\  { A ,  v }  e.  E )  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
5453ex 436 . . . . . . . . . . . . . 14  |-  ( G  e. USGraph  ->  ( { A ,  v }  e.  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5510, 54syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( { A , 
v }  e.  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5655adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) ) )
5756com3r 82 . . . . . . . . . . 11  |-  ( v  =  A  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
58 orc 387 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
v  =  B  \/  v  =  C )
)
59582a1d 27 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
60 olc 386 . . . . . . . . . . . 12  |-  ( v  =  C  ->  (
v  =  B  \/  v  =  C )
)
61602a1d 27 . . . . . . . . . . 11  |-  ( v  =  C  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
6257, 59, 613jaoi 1331 . . . . . . . . . 10  |-  ( ( v  =  A  \/  v  =  B  \/  v  =  C )  ->  ( ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E ) )  -> 
( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6346, 62sylbi 199 . . . . . . . . 9  |-  ( v  e.  { A ,  B ,  C }  ->  ( ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E ) )  -> 
( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6463com12 32 . . . . . . . 8  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  { A ,  B ,  C }  ->  ( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6544, 64sylbid 219 . . . . . . 7  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  -> 
( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6665impd 433 . . . . . 6  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  E )  ->  ( v  =  B  \/  v  =  C ) ) )
67 eqid 2451 . . . . . . . . . . . . . . . . . 18  |-  B  =  B
68673mix2i 1181 . . . . . . . . . . . . . . . . 17  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
691simp2d 1021 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  B  e.  Y )
70 eltpg 4014 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
7169, 70syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C ) ) )
7268, 71mpbiri 237 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
7372adantr 467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  B )  ->  B  e.  { A ,  B ,  C } )
74 eleq1 2517 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7574bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7675adantl 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  B )  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7773, 76mpbid 214 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  B )  ->  v  e.  { A ,  B ,  C } )
7842bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
7941, 78syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
8079adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  B )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
8177, 80mpbid 214 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  =  B )  ->  v  e.  V )
8281ex 436 . . . . . . . . . . . 12  |-  ( ph  ->  ( v  =  B  ->  v  e.  V
) )
8382adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  B  -> 
v  e.  V ) )
8483impcom 432 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
v  e.  V )
85 preq2 4052 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8685eleq1d 2513 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  E  <->  { A ,  v }  e.  E ) )
8786eqcoms 2459 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( { A ,  B }  e.  E  <->  { A ,  v }  e.  E ) )
8887biimpcd 228 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  E  ->  ( v  =  B  ->  { A ,  v }  e.  E ) )
8988ad2antrl 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  B  ->  { A ,  v }  e.  E ) )
9089impcom 432 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  ->  { A ,  v }  e.  E )
9184, 90jca 535 . . . . . . . . 9  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) )
9291ex 436 . . . . . . . 8  |-  ( v  =  B  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  /\  { A ,  v }  e.  E ) ) )
93 tpid3g 4087 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
94933ad2ant3 1031 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { A ,  B ,  C }
)
951, 94syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  C  e.  { A ,  B ,  C }
)
9695adantr 467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  C )  ->  C  e.  { A ,  B ,  C } )
97 eleq1 2517 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9897bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9998adantl 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  C )  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
10096, 99mpbid 214 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  C )  ->  v  e.  { A ,  B ,  C } )
10179adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  C )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
102100, 101mpbid 214 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  =  C )  ->  v  e.  V )
103102ex 436 . . . . . . . . . . . 12  |-  ( ph  ->  ( v  =  C  ->  v  e.  V
) )
104103adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  C  -> 
v  e.  V ) )
105104impcom 432 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
v  e.  V )
106 preq2 4052 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
107106eleq1d 2513 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  E  <->  { A ,  v }  e.  E ) )
108107eqcoms 2459 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  ( { A ,  C }  e.  E  <->  { A ,  v }  e.  E ) )
109108biimpcd 228 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  E  ->  ( v  =  C  ->  { A ,  v }  e.  E ) )
110109ad2antll 735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  C  ->  { A ,  v }  e.  E ) )
111110impcom 432 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  ->  { A ,  v }  e.  E )
112105, 111jca 535 . . . . . . . . 9  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) )
113112ex 436 . . . . . . . 8  |-  ( v  =  C  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  /\  { A ,  v }  e.  E ) ) )
11492, 113jaoi 381 . . . . . . 7  |-  ( ( v  =  B  \/  v  =  C )  ->  ( ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) ) )
115114com12 32 . . . . . 6  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  =  B  \/  v  =  C )  ->  ( v  e.  V  /\  { A ,  v }  e.  E ) ) )
11666, 115impbid 194 . . . . 5  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  E )  <-> 
( v  =  B  \/  v  =  C ) ) )
117116abbidv 2569 . . . 4  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  { v  |  ( v  e.  V  /\  { A ,  v }  e.  E ) }  =  { v  |  ( v  =  B  \/  v  =  C ) } )
118 df-rab 2746 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  E }  =  {
v  |  ( v  e.  V  /\  { A ,  v }  e.  E ) }
119 dfpr2 3983 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
120117, 118, 1193eqtr4g 2510 . . 3  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  { v  e.  V  |  { A ,  v }  e.  E }  =  { B ,  C }
)
12140, 120eqtrd 2485 . 2  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( G NeighbVtx  A )  =  { B ,  C }
)
12236, 121impbida 843 1  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    \/ w3o 984    /\ w3a 985    = wceq 1444    e. wcel 1887   {cab 2437    =/= wne 2622   {crab 2741   {cpr 3970   {ctp 3972   ` cfv 5582  (class class class)co 6290  Vtxcvtx 39101  Edgcedga 39210   USGraph cusgr 39236   NeighbVtx cnbgr 39397
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-fal 1450  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-2o 7183  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-upgr 39174  df-umgr 39175  df-edga 39211  df-usgr 39238  df-nbgr 39401
This theorem is referenced by:  nb3grpr  39456  nb3grpr2  39457  nb3gr2nb  39458
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