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Theorem nb3graprlem2 24579
Description: Lemma 2 for nb3grapr 24580. (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 4042 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
21difeq2d 3618 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
3 preq1 4111 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
43eqeq2d 2471 . . . . 5  |-  ( v  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  w } ) )
52, 4rexeqbidv 3069 . . . 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 4042 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
76difeq2d 3618 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
8 preq1 4111 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
98eqeq2d 2471 . . . . 5  |-  ( v  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  w } ) )
107, 9rexeqbidv 3069 . . . 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 4042 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1211difeq2d 3618 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
13 preq1 4111 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1413eqeq2d 2471 . . . . 5  |-  ( v  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  w } ) )
1512, 14rexeqbidv 3069 . . . 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 4084 . . 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 1017 . 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 457 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  V  =  { A ,  B ,  C }
)
19 difeq1 3611 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  ( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2019adantr 465 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2120rexeqdv 3061 . . . 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 3069 . . 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 1018 . 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 4112 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2524eqeq2d 2471 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  B } ) )
26 preq2 4112 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
2726eqeq2d 2471 . . . . . . 7  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  C } ) )
2825, 27rexprg 4082 . . . . . 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 1014 . . . . 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 4112 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3130eqeq2d 2471 . . . . . . . 8  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  C } ) )
32 preq2 4112 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3332eqeq2d 2471 . . . . . . . 8  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
3431, 33rexprg 4082 . . . . . . 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 453 . . . . . 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 1015 . . . . 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 4112 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
3837eqeq2d 2471 . . . . . . 7  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
39 preq2 4112 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4039eqeq2d 2471 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  B } ) )
4138, 40rexprg 4082 . . . . . 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 1016 . . . . 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 1298 . . . 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 1017 . . 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 4127 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
4645a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
4746difeq1d 3617 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
48 necom 2726 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
49 necom 2726 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
50 diftpsn3 4170 . . . . . . . . 9  |-  ( ( B  =/=  A  /\  C  =/=  A )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
5148, 49, 50syl2anb 479 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
52513adant3 1016 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5347, 52eqtrd 2498 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5453rexeqdv 3061 . . . . 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 4127 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
5655eqcomi 2470 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
5756a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
5857difeq1d 3617 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
59 necom 2726 . . . . . . . . . . . 12  |-  ( B  =/=  C  <->  C  =/=  B )
6059anbi1i 695 . . . . . . . . . . 11  |-  ( ( B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  B  /\  A  =/= 
B ) )
6160biimpi 194 . . . . . . . . . 10  |-  ( ( B  =/=  C  /\  A  =/=  B )  -> 
( C  =/=  B  /\  A  =/=  B
) )
6261ancoms 453 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( C  =/=  B  /\  A  =/=  B
) )
63 diftpsn3 4170 . . . . . . . . 9  |-  ( ( C  =/=  B  /\  A  =/=  B )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
6462, 63syl 16 . . . . . . . 8  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
65643adant2 1015 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
6658, 65eqtrd 2498 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
6766rexeqdv 3061 . . . . 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 4170 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
69683adant1 1014 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7069rexeqdv 3061 . . . . 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 1298 . . . 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 1019 . . 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 4110 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7473eqeq2i 2475 . . . . . . 7  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  B }  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C } )
7574orbi2i 519 . . . . . 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 514 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)
7775, 76bitr2i 250 . . . . 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 24563 . . . . . . . . . 10  |-  ( V USGrph  E  ->  A  e/  ( <. V ,  E >. Neighbors  A
) )
80 df-nel 2655 . . . . . . . . . . 11  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  <->  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )
81 prid2g 4139 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
82813ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
83 eleq2 2530 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { B ,  A } ) )
8482, 83syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
8584con3rr3 136 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8680, 85sylbi 195 . . . . . . . . . 10  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( <. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
8779, 86syl 16 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8887adantl 466 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8988impcom 430 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)
90893adant3 1016 . . . . . 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 405 . . . . . . 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 387 . . . . . . 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 261 . . . . . 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 16 . . . . 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 4139 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
96953ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
97 eleq2 2530 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { C ,  A } ) )
9896, 97syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
9998con3rr3 136 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
10080, 99sylbi 195 . . . . . . . . . 10  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( <. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
10179, 100syl 16 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
102101adantl 466 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
103102impcom 430 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
)
1041033adant3 1016 . . . . . 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 405 . . . . . 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 16 . . . . 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 709 . . . 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 4138 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1091083ad2ant1 1017 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
110 eleq2 2530 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  B }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  B } ) )
111109, 110syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
112111con3dimp 441 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }
)
113 prid1g 4138 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1141133ad2ant1 1017 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
115 eleq2 2530 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  C }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  C } ) )
116114, 115syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  C }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
117116con3dimp 441 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)
118112, 117jca 532 . . . . . . . . . . . 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 435 . . . . . . . . . . 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 195 . . . . . . . . . 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 16 . . . . . . . . 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 466 . . . . . . . 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 430 . . . . . . 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 1016 . . . . . 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 490 . . . . . 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 212 . . . . 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 1334 . . . 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 279 . . 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 286 . 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 286 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 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    e/ wnel 2653   E.wrex 2808    \ cdif 3468   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547
This theorem is referenced by:  nb3grapr  24580
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