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Theorem nb3grprlem2 39465
Description: Lemma 2 for nb3grapr 25193. (Contributed by Alexander van der Vekens, 17-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
) )
nb3grpr.n  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
Assertion
Ref Expression
nb3grprlem2  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } ) )
Distinct variable groups:    v, A    v, B    v, C    v, E    v, G    v, V    ph, v    w, A, v   
w, B    w, C    w, G    w, V
Allowed substitution hints:    ph( w)    E( w)    X( w, v)    Y( w, v)    Z( w, v)

Proof of Theorem nb3grprlem2
StepHypRef Expression
1 nb3grpr.s . . 3  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
2 sneq 3980 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
32difeq2d 3553 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
4 preq1 4054 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
54eqeq2d 2463 . . . . 5  |-  ( v  =  A  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  w } ) )
63, 5rexeqbidv 3004 . . . 4  |-  ( v  =  A  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }
) )
7 sneq 3980 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
87difeq2d 3553 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
9 preq1 4054 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
109eqeq2d 2463 . . . . 5  |-  ( v  =  B  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  w } ) )
118, 10rexeqbidv 3004 . . . 4  |-  ( v  =  B  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }
) )
12 sneq 3980 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1312difeq2d 3553 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
14 preq1 4054 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1514eqeq2d 2463 . . . . 5  |-  ( v  =  C  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  w } ) )
1613, 15rexeqbidv 3004 . . . 4  |-  ( v  =  C  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w }
) )
176, 11, 16rextpg 4026 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } 
<->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
181, 17syl 17 . 2  |-  ( ph  ->  ( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } 
<->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
19 nb3grpr.t . . . 4  |-  ( ph  ->  V  =  { A ,  B ,  C }
)
20 nb3grpr.g . . . 4  |-  ( ph  ->  G  e. USGraph  )
2119, 20jca 535 . . 3  |-  ( ph  ->  ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )
22 simpl 459 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  ->  V  =  { A ,  B ,  C }
)
23 difeq1 3546 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  ( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2423adantr 467 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2524rexeqdv 2996 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
2622, 25rexeqbidv 3004 . . 3  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( E. v  e.  V  E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
2721, 26syl 17 . 2  |-  ( ph  ->  ( E. v  e.  V  E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
28 preq2 4055 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2928eqeq2d 2463 . . . . . . 7  |-  ( w  =  B  ->  (
( G NeighbVtx  A )  =  { A ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  B } ) )
30 preq2 4055 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
3130eqeq2d 2463 . . . . . . 7  |-  ( w  =  C  ->  (
( G NeighbVtx  A )  =  { A ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  C } ) )
3229, 31rexprg 4024 . . . . . 6  |-  ( ( B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  <->  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) ) )
33323adant1 1027 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  <->  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) ) )
34 preq2 4055 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3534eqeq2d 2463 . . . . . . . 8  |-  ( w  =  C  ->  (
( G NeighbVtx  A )  =  { B ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  C } ) )
36 preq2 4055 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3736eqeq2d 2463 . . . . . . . 8  |-  ( w  =  A  ->  (
( G NeighbVtx  A )  =  { B ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  A } ) )
3835, 37rexprg 4024 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
3938ancoms 455 . . . . . 6  |-  ( ( A  e.  X  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
40393adant2 1028 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
41 preq2 4055 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
4241eqeq2d 2463 . . . . . . 7  |-  ( w  =  A  ->  (
( G NeighbVtx  A )  =  { C ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  A } ) )
43 preq2 4055 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4443eqeq2d 2463 . . . . . . 7  |-  ( w  =  B  ->  (
( G NeighbVtx  A )  =  { C ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  B } ) )
4542, 44rexprg 4024 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
46453adant3 1029 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
4733, 40, 463orbi123d 1340 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( (
( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A }
)  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) ) )
481, 47syl 17 . . 3  |-  ( ph  ->  ( ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( (
( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A }
)  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) ) )
49 nb3grpr.n . . . 4  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
50 tprot 4070 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
5150a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
5251difeq1d 3552 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
53 necom 2679 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
54 necom 2679 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
55 diftpsn3 4113 . . . . . . . . 9  |-  ( ( B  =/=  A  /\  C  =/=  A )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
5653, 54, 55syl2anb 482 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
57563adant3 1029 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5852, 57eqtrd 2487 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5958rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  <->  E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }
) )
60 tprot 4070 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
6160eqcomi 2462 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
6261a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
6362difeq1d 3552 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
64 necom 2679 . . . . . . . . . . . 12  |-  ( B  =/=  C  <->  C  =/=  B )
6564anbi1i 702 . . . . . . . . . . 11  |-  ( ( B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  B  /\  A  =/= 
B ) )
6665biimpi 198 . . . . . . . . . 10  |-  ( ( B  =/=  C  /\  A  =/=  B )  -> 
( C  =/=  B  /\  A  =/=  B
) )
6766ancoms 455 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( C  =/=  B  /\  A  =/=  B
) )
68 diftpsn3 4113 . . . . . . . . 9  |-  ( ( C  =/=  B  /\  A  =/=  B )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
6967, 68syl 17 . . . . . . . 8  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
70693adant2 1028 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
7163, 70eqtrd 2487 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
7271rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  <->  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }
) )
73 diftpsn3 4113 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
74733adant1 1027 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7574rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w }  <->  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }
) )
7659, 72, 753orbi123d 1340 . . . 4  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  (
( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } ) ) )
7749, 76syl 17 . . 3  |-  ( ph  ->  ( ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } ) ) )
78 prcom 4053 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7978eqeq2i 2465 . . . . . . 7  |-  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( G NeighbVtx  A )  =  { B ,  C }
)
8079orbi2i 522 . . . . . 6  |-  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  C } ) )
81 oridm 517 . . . . . 6  |-  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  C } )  <->  ( G NeighbVtx  A )  =  { B ,  C } )
8280, 81bitr2i 254 . . . . 5  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) )
8382a1i 11 . . . 4  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
84 usgrnbnself2 39444 . . . . . . . 8  |-  ( G  e. USGraph  ->  A  e/  ( G NeighbVtx  A ) )
85 df-nel 2627 . . . . . . . . 9  |-  ( A  e/  ( G NeighbVtx  A )  <->  -.  A  e.  ( G NeighbVtx  A ) )
86 prid2g 4082 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
87863ad2ant1 1030 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
88 eleq2 2520 . . . . . . . . . . 11  |-  ( ( G NeighbVtx  A )  =  { B ,  A }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { B ,  A }
) )
8987, 88syl5ibrcom 226 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { B ,  A }  ->  A  e.  ( G NeighbVtx  A )
) )
9089con3rr3 142 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( G NeighbVtx  A )  =  { B ,  A }
) )
9185, 90sylbi 199 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } ) )
9284, 91syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } ) )
9320, 1, 92sylc 62 . . . . . 6  |-  ( ph  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } )
94 biorf 407 . . . . . . 7  |-  ( -.  ( G NeighbVtx  A )  =  { B ,  A }  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  A }  \/  ( G NeighbVtx  A )  =  { B ,  C } ) ) )
95 orcom 389 . . . . . . 7  |-  ( ( ( G NeighbVtx  A )  =  { B ,  A }  \/  ( G NeighbVtx  A )  =  { B ,  C } )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) )
9694, 95syl6bb 265 . . . . . 6  |-  ( -.  ( G NeighbVtx  A )  =  { B ,  A }  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
9793, 96syl 17 . . . . 5  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
98 prid2g 4082 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
99983ad2ant1 1030 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
100 eleq2 2520 . . . . . . . . . . 11  |-  ( ( G NeighbVtx  A )  =  { C ,  A }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { C ,  A }
) )
10199, 100syl5ibrcom 226 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { C ,  A }  ->  A  e.  ( G NeighbVtx  A )
) )
102101con3rr3 142 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( G NeighbVtx  A )  =  { C ,  A }
) )
10385, 102sylbi 199 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } ) )
10484, 103syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } ) )
10520, 1, 104sylc 62 . . . . . 6  |-  ( ph  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } )
106 biorf 407 . . . . . 6  |-  ( -.  ( G NeighbVtx  A )  =  { C ,  A }  ->  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
107105, 106syl 17 . . . . 5  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
10897, 107orbi12d 717 . . . 4  |-  ( ph  ->  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } )  <->  ( (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
109 prid1g 4081 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1101093ad2ant1 1030 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
111 eleq2 2520 . . . . . . . . . . . . 13  |-  ( ( G NeighbVtx  A )  =  { A ,  B }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { A ,  B }
) )
112110, 111syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { A ,  B }  ->  A  e.  ( G NeighbVtx  A )
) )
113112con3dimp 443 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  -.  ( G NeighbVtx  A )  =  { A ,  B }
)
114 prid1g 4081 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1151143ad2ant1 1030 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
116 eleq2 2520 . . . . . . . . . . . . 13  |-  ( ( G NeighbVtx  A )  =  { A ,  C }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { A ,  C }
) )
117115, 116syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { A ,  C }  ->  A  e.  ( G NeighbVtx  A )
) )
118117con3dimp 443 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  -.  ( G NeighbVtx  A )  =  { A ,  C }
)
119113, 118jca 535 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) )
120119expcom 437 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12185, 120sylbi 199 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12284, 121syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12320, 1, 122sylc 62 . . . . . 6  |-  ( ph  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C } ) )
124 ioran 493 . . . . . 6  |-  ( -.  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C }
)  <->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C } ) )
125123, 124sylibr 216 . . . . 5  |-  ( ph  ->  -.  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) )
1261253bior1fd 1377 . . . 4  |-  ( ph  ->  ( ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  (
( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) )  <-> 
( ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
12783, 108, 1263bitrd 283 . . 3  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
12848, 77, 1273bitr4rd 290 . 2  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
12918, 27, 1283bitr4rd 290 1  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    \/ w3o 985    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624    e/ wnel 2625   E.wrex 2740    \ cdif 3403   {csn 3970   {cpr 3972   {ctp 3974   ` cfv 5585  (class class class)co 6295  Vtxcvtx 39111  Edgcedga 39220   USGraph cusgr 39246   NeighbVtx cnbgr 39407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-nbgr 39411
This theorem is referenced by:  nb3grpr  39466
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