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Theorem nb3grprlem1 39618
Description: Lemma 1 for nb3grapr 25260. (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 4069 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
323ad2ant2 1052 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
41, 3syl 17 . . . . . 6  |-  ( ph  ->  B  e.  { B ,  C } )
54adantr 472 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  B  e.  { B ,  C } )
6 eleq2 2538 . . . . . . 7  |-  ( { B ,  C }  =  ( G NeighbVtx  A )  ->  ( B  e. 
{ B ,  C } 
<->  B  e.  ( G NeighbVtx  A ) ) )
76eqcoms 2479 . . . . . 6  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  ->  ( B  e.  { B ,  C }  <->  B  e.  ( G NeighbVtx  A ) ) )
87adantl 473 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( B  e.  { B ,  C }  <->  B  e.  ( G NeighbVtx  A ) ) )
95, 8mpbid 215 . . . 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 39585 . . . . . . 7  |-  ( G  e. USGraph  ->  ( B  e.  ( G NeighbVtx  A )  <->  { B ,  A }  e.  E ) )
13 prcom 4041 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1413a1i 11 . . . . . . . 8  |-  ( G  e. USGraph  ->  { B ,  A }  =  { A ,  B }
)
1514eleq1d 2533 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { B ,  A }  e.  E  <->  { A ,  B }  e.  E ) )
1612, 15bitrd 261 . . . . . 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 472 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( B  e.  ( G NeighbVtx  A )  <->  { A ,  B }  e.  E
) )
199, 18mpbid 215 . . 3  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  { A ,  B }  e.  E )
20 prid2g 4070 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
21203ad2ant3 1053 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
221, 21syl 17 . . . . . 6  |-  ( ph  ->  C  e.  { B ,  C } )
2322adantr 472 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  C  e.  { B ,  C } )
24 eleq2 2538 . . . . . . 7  |-  ( { B ,  C }  =  ( G NeighbVtx  A )  ->  ( C  e. 
{ B ,  C } 
<->  C  e.  ( G NeighbVtx  A ) ) )
2524eqcoms 2479 . . . . . 6  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  ->  ( C  e.  { B ,  C }  <->  C  e.  ( G NeighbVtx  A ) ) )
2625adantl 473 . . . . 5  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( C  e.  { B ,  C }  <->  C  e.  ( G NeighbVtx  A ) ) )
2723, 26mpbid 215 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  C  e.  ( G NeighbVtx  A ) )
2811nbusgreledg 39585 . . . . . . 7  |-  ( G  e. USGraph  ->  ( C  e.  ( G NeighbVtx  A )  <->  { C ,  A }  e.  E ) )
29 prcom 4041 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
3029a1i 11 . . . . . . . 8  |-  ( G  e. USGraph  ->  { C ,  A }  =  { A ,  C }
)
3130eleq1d 2533 . . . . . . 7  |-  ( G  e. USGraph  ->  ( { C ,  A }  e.  E  <->  { A ,  C }  e.  E ) )
3228, 31bitrd 261 . . . . . 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 472 . . . 4  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( C  e.  ( G NeighbVtx  A )  <->  { A ,  C }  e.  E
) )
3527, 34mpbid 215 . . 3  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  ->  { A ,  C }  e.  E )
3619, 35jca 541 . 2  |-  ( (
ph  /\  ( G NeighbVtx  A )  =  { B ,  C } )  -> 
( { A ,  B }  e.  E  /\  { A ,  C }  e.  E )
)
37 nb3grpr.v . . . . . 6  |-  V  =  (Vtx `  G )
3837, 11nbusgr 39581 . . . . 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 472 . . 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 2538 . . . . . . . . . 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 472 . . . . . . . 8  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
45 vex 3034 . . . . . . . . . . 11  |-  v  e. 
_V
4645eltp 4008 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
4711usgredgne 39451 . . . . . . . . . . . . . . . 16  |-  ( ( G  e. USGraph  /\  { A ,  v }  e.  E )  ->  A  =/=  v )
48 df-ne 2643 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
49 pm2.24 112 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
5049eqcoms 2479 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  A  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
5150com12 31 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  =  v  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) )
5248, 51sylbi 200 . . . . . . . . . . . . . . . 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 441 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) ) )
5756com3r 81 . . . . . . . . . . 11  |-  ( v  =  A  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
58 orc 392 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
v  =  B  \/  v  =  C )
)
59582a1d 26 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
60 olc 391 . . . . . . . . . . . 12  |-  ( v  =  C  ->  (
v  =  B  \/  v  =  C )
)
61602a1d 26 . . . . . . . . . . 11  |-  ( v  =  C  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( { A ,  v }  e.  E  ->  (
v  =  B  \/  v  =  C )
) ) )
6257, 59, 613jaoi 1357 . . . . . . . . . 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 200 . . . . . . . . 9  |-  ( v  e.  { A ,  B ,  C }  ->  ( ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E ) )  -> 
( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6463com12 31 . . . . . . . 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 223 . . . . . . 7  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  -> 
( { A , 
v }  e.  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6665impd 438 . . . . . 6  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  E )  ->  ( v  =  B  \/  v  =  C ) ) )
67 eqid 2471 . . . . . . . . . . . . . . . . . 18  |-  B  =  B
68673mix2i 1203 . . . . . . . . . . . . . . . . 17  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
691simp2d 1043 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  B  e.  Y )
70 eltpg 4005 . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
7372adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  B )  ->  B  e.  { A ,  B ,  C } )
74 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7574bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7675adantl 473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  B )  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7773, 76mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  B )  ->  v  e.  { A ,  B ,  C } )
7842bicomd 206 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  B )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
8177, 80mpbid 215 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  =  B )  ->  v  e.  V )
8281ex 441 . . . . . . . . . . . 12  |-  ( ph  ->  ( v  =  B  ->  v  e.  V
) )
8382adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  B  -> 
v  e.  V ) )
8483impcom 437 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
v  e.  V )
85 preq2 4043 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8685eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  E  <->  { A ,  v }  e.  E ) )
8786eqcoms 2479 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( { A ,  B }  e.  E  <->  { A ,  v }  e.  E ) )
8887biimpcd 232 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  E  ->  ( v  =  B  ->  { A ,  v }  e.  E ) )
8988ad2antrl 742 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  B  ->  { A ,  v }  e.  E ) )
9089impcom 437 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  ->  { A ,  v }  e.  E )
9184, 90jca 541 . . . . . . . . 9  |-  ( ( v  =  B  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) )
9291ex 441 . . . . . . . 8  |-  ( v  =  B  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  /\  { A ,  v }  e.  E ) ) )
93 tpid3g 4078 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
94933ad2ant3 1053 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  C )  ->  C  e.  { A ,  B ,  C } )
97 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9897bicomd 206 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9998adantl 473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  v  =  C )  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
10096, 99mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  C )  ->  v  e.  { A ,  B ,  C } )
10179adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  =  C )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
102100, 101mpbid 215 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  =  C )  ->  v  e.  V )
103102ex 441 . . . . . . . . . . . 12  |-  ( ph  ->  ( v  =  C  ->  v  e.  V
) )
104103adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  C  -> 
v  e.  V ) )
105104impcom 437 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
v  e.  V )
106 preq2 4043 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
107106eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  E  <->  { A ,  v }  e.  E ) )
108107eqcoms 2479 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  ( { A ,  C }  e.  E  <->  { A ,  v }  e.  E ) )
109108biimpcd 232 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  E  ->  ( v  =  C  ->  { A ,  v }  e.  E ) )
110109ad2antll 743 . . . . . . . . . . 11  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  =  C  ->  { A ,  v }  e.  E ) )
111110impcom 437 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  ->  { A ,  v }  e.  E )
112105, 111jca 541 . . . . . . . . 9  |-  ( ( v  =  C  /\  ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) )
113112ex 441 . . . . . . . 8  |-  ( v  =  C  ->  (
( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
v  e.  V  /\  { A ,  v }  e.  E ) ) )
11492, 113jaoi 386 . . . . . . 7  |-  ( ( v  =  B  \/  v  =  C )  ->  ( ( ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  E ) ) )
115114com12 31 . . . . . 6  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  =  B  \/  v  =  C )  ->  ( v  e.  V  /\  { A ,  v }  e.  E ) ) )
11666, 115impbid 195 . . . . 5  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  E )  <-> 
( v  =  B  \/  v  =  C ) ) )
117116abbidv 2589 . . . 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 2765 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  E }  =  {
v  |  ( v  e.  V  /\  { A ,  v }  e.  E ) }
119 dfpr2 3974 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
120117, 118, 1193eqtr4g 2530 . . 3  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  { v  e.  V  |  { A ,  v }  e.  E }  =  { B ,  C }
)
12140, 120eqtrd 2505 . 2  |-  ( (
ph  /\  ( { A ,  B }  e.  E  /\  { A ,  C }  e.  E
) )  ->  ( G NeighbVtx  A )  =  { B ,  C }
)
12236, 121impbida 850 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 189    \/ wo 375    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   {crab 2760   {cpr 3961   {ctp 3963   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251  Edgcedga 39371   USGraph cusgr 39397   NeighbVtx cnbgr 39561
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-fal 1458  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-2o 7201  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-upgr 39328  df-umgr 39329  df-edga 39372  df-usgr 39399  df-nbgr 39565
This theorem is referenced by:  nb3grpr  39620  nb3grpr2  39621  nb3gr2nb  39622
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