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Theorem nb3graprlem2 25192
Description: Lemma 2 for nb3grapr 25193. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
Assertion
Ref Expression
nb3graprlem2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }
) )
Distinct variable groups:    v, E, w    v, V, w    v, A, w    v, B, w   
v, C, w
Allowed substitution hints:    X( w, v)    Y( w, v)    Z( w, v)

Proof of Theorem nb3graprlem2
StepHypRef Expression
1 sneq 3980 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
21difeq2d 3553 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
3 preq1 4054 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
43eqeq2d 2463 . . . . 5  |-  ( v  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  w } ) )
52, 4rexeqbidv 3004 . . . 4  |-  ( v  =  A  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A )  =  { A ,  w }
) )
6 sneq 3980 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
76difeq2d 3553 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
8 preq1 4054 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
98eqeq2d 2463 . . . . 5  |-  ( v  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  w } ) )
107, 9rexeqbidv 3004 . . . 4  |-  ( v  =  B  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A )  =  { B ,  w }
) )
11 sneq 3980 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1211difeq2d 3553 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
13 preq1 4054 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1413eqeq2d 2463 . . . . 5  |-  ( v  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  w } ) )
1512, 14rexeqbidv 3004 . . . 4  |-  ( v  =  C  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A )  =  { C ,  w }
) )
165, 10, 15rextpg 4026 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } ) ) )
17163ad2ant1 1030 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } ) ) )
18 simpl 459 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  V  =  { A ,  B ,  C }
)
19 difeq1 3546 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  ( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2019adantr 467 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2120rexeqdv 2996 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( E. w  e.  ( V  \  {
v } ) (
<. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }
) )
2218, 21rexeqbidv 3004 . . 3  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( E. v  e.  V  E. w  e.  ( V  \  {
v } ) (
<. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }
) )
23223ad2ant2 1031 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( E. v  e.  V  E. w  e.  ( V  \  {
v } ) (
<. V ,  E >. Neighbors  A
)  =  { v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( <. V ,  E >. Neighbors  A )  =  {
v ,  w }
) )
24 preq2 4055 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2524eqeq2d 2463 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  B } ) )
26 preq2 4055 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
2726eqeq2d 2463 . . . . . . 7  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  C } ) )
2825, 27rexprg 4024 . . . . . 6  |-  ( ( B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
29283adant1 1027 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
30 preq2 4055 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3130eqeq2d 2463 . . . . . . . 8  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  C } ) )
32 preq2 4055 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3332eqeq2d 2463 . . . . . . . 8  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
3431, 33rexprg 4024 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X )  ->  ( E. w  e. 
{ C ,  A }  ( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) ) )
3534ancoms 455 . . . . . 6  |-  ( ( A  e.  X  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) ) )
36353adant2 1028 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) ) )
37 preq2 4055 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
3837eqeq2d 2463 . . . . . . 7  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
39 preq2 4055 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4039eqeq2d 2463 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  B } ) )
4138, 40rexprg 4024 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  ( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) )
42413adant3 1029 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ A ,  B }  ( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) )
4329, 36, 423orbi123d 1340 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( E. w  e.  { B ,  C }  ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B } 
( <. V ,  E >. Neighbors  A )  =  { C ,  w }
)  <->  ( ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A
)  =  { A ,  C } )  \/  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)  \/  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A
)  =  { C ,  B } ) ) ) )
44433ad2ant1 1030 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( E. w  e.  { B ,  C }  ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B } 
( <. V ,  E >. Neighbors  A )  =  { C ,  w }
)  <->  ( ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A
)  =  { A ,  C } )  \/  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)  \/  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A
)  =  { C ,  B } ) ) ) )
45 tprot 4070 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
4645a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
4746difeq1d 3552 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
48 necom 2679 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
49 necom 2679 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
50 diftpsn3 4113 . . . . . . . . 9  |-  ( ( B  =/=  A  /\  C  =/=  A )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
5148, 49, 50syl2anb 482 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
52513adant3 1029 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5347, 52eqtrd 2487 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5453rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  <->  E. w  e.  { B ,  C }  ( <. V ,  E >. Neighbors  A )  =  { A ,  w }
) )
55 tprot 4070 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
5655eqcomi 2462 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
5756a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
5857difeq1d 3552 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
59 necom 2679 . . . . . . . . . . . 12  |-  ( B  =/=  C  <->  C  =/=  B )
6059anbi1i 702 . . . . . . . . . . 11  |-  ( ( B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  B  /\  A  =/= 
B ) )
6160biimpi 198 . . . . . . . . . 10  |-  ( ( B  =/=  C  /\  A  =/=  B )  -> 
( C  =/=  B  /\  A  =/=  B
) )
6261ancoms 455 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( C  =/=  B  /\  A  =/=  B
) )
63 diftpsn3 4113 . . . . . . . . 9  |-  ( ( C  =/=  B  /\  A  =/=  B )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
6462, 63syl 17 . . . . . . . 8  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
65643adant2 1028 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
6658, 65eqtrd 2487 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
6766rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  <->  E. w  e.  { C ,  A }  ( <. V ,  E >. Neighbors  A )  =  { B ,  w }
) )
68 diftpsn3 4113 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
69683adant1 1027 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7069rexeqdv 2996 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w }  <->  E. w  e.  { A ,  B }  ( <. V ,  E >. Neighbors  A )  =  { C ,  w }
) )
7154, 67, 703orbi123d 1340 . . . 4  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  (
( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } ) ) )
72713ad2ant3 1032 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } ) ) )
73 prcom 4053 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7473eqeq2i 2465 . . . . . . 7  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  B }  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C } )
7574orbi2i 522 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
)  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { B ,  C } ) )
76 oridm 517 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)
7775, 76bitr2i 254 . . . . 5  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { C ,  B } ) )
7877a1i 11 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) )
79 nbgranself2 25176 . . . . . . . . . 10  |-  ( V USGrph  E  ->  A  e/  ( <. V ,  E >. Neighbors  A
) )
80 df-nel 2627 . . . . . . . . . . 11  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  <->  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )
81 prid2g 4082 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
82813ad2ant1 1030 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
83 eleq2 2520 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { B ,  A } ) )
8482, 83syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
8584con3rr3 142 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8680, 85sylbi 199 . . . . . . . . . 10  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
8779, 86syl 17 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8887adantl 468 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8988impcom 432 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)
90893adant3 1029 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)
91 biorf 407 . . . . . . 7  |-  ( -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
) ) )
92 orcom 389 . . . . . . 7  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
9391, 92syl6bb 265 . . . . . 6  |-  ( -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) ) )
9490, 93syl 17 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) ) )
95 prid2g 4082 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
96953ad2ant1 1030 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
97 eleq2 2520 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { C ,  A } ) )
9896, 97syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
9998con3rr3 142 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
10080, 99sylbi 199 . . . . . . . . . 10  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( <. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
10179, 100syl 17 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
102101adantl 468 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
103102impcom 432 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
)
1041033adant3 1029 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
)
105 biorf 407 . . . . . 6  |-  ( -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  ->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  B }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) )
106104, 105syl 17 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { C ,  B }  <->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) )
10794, 106orbi12d 717 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { C ,  B } )  <->  ( (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)  \/  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A
)  =  { C ,  B } ) ) ) )
108 prid1g 4081 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1091083ad2ant1 1030 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
110 eleq2 2520 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  B }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  B } ) )
111109, 110syl5ibrcom 226 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
112111con3dimp 443 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }
)
113 prid1g 4081 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1141133ad2ant1 1030 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
115 eleq2 2520 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  C }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  C } ) )
116114, 115syl5ibrcom 226 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  C }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
117116con3dimp 443 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)
118112, 117jca 535 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  -> 
( -.  ( <. V ,  E >. Neighbors  A
)  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) )
119118expcom 437 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( <. V ,  E >. Neighbors  A
)  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
12080, 119sylbi 199 . . . . . . . . . 10  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  ( -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
12179, 120syl 17 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
122121adantl 468 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) ) )
123122impcom 432 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) )
1241233adant3 1029 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( -.  ( <. V ,  E >. Neighbors  A
)  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) )
125 ioran 493 . . . . . 6  |-  ( -.  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)  <->  ( -.  ( <. V ,  E >. Neighbors  A
)  =  { A ,  B }  /\  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) )
126124, 125sylibr 216 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  -.  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
) )
1271263bior1fd 1377 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } )  \/  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) )  <->  ( (
( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)  \/  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } )  \/  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) ) )
12878, 107, 1273bitrd 283 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  \/  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)  \/  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } )  \/  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  \/  ( <. V ,  E >. Neighbors  A )  =  { C ,  B }
) ) ) )
12944, 72, 1283bitr4rd 290 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( <. V ,  E >. Neighbors  A
)  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( <. V ,  E >. Neighbors  A
)  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( <. V ,  E >. Neighbors  A
)  =  { C ,  w } ) ) )
13017, 23, 1293bitr4rd 290 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( <. V ,  E >. Neighbors  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   <.cop 3976   class class class wbr 4405  (class class class)co 6295   USGrph cusg 25069   Neighbors cnbgra 25157
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-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  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-reu 2746  df-rmo 2747  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-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-usgra 25072  df-nbgra 25160
This theorem is referenced by:  nb3grapr  25193
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