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Theorem nb3graprlem2 25165
Description: Lemma 2 for nb3grapr 25166. (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 4006 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
21difeq2d 3583 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
3 preq1 4076 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
43eqeq2d 2436 . . . . 5  |-  ( v  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  w } ) )
52, 4rexeqbidv 3040 . . . 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 4006 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
76difeq2d 3583 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
8 preq1 4076 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
98eqeq2d 2436 . . . . 5  |-  ( v  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  w } ) )
107, 9rexeqbidv 3040 . . . 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 4006 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1211difeq2d 3583 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
13 preq1 4076 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1413eqeq2d 2436 . . . . 5  |-  ( v  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  w } ) )
1512, 14rexeqbidv 3040 . . . 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 4049 . . 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 1026 . 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 458 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  V  =  { A ,  B ,  C }
)
19 difeq1 3576 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  ( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2019adantr 466 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2120rexeqdv 3032 . . . 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 3040 . . 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 1027 . 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 4077 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2524eqeq2d 2436 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  B } ) )
26 preq2 4077 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
2726eqeq2d 2436 . . . . . . 7  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  C } ) )
2825, 27rexprg 4047 . . . . . 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 1023 . . . . 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 4077 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3130eqeq2d 2436 . . . . . . . 8  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  C } ) )
32 preq2 4077 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3332eqeq2d 2436 . . . . . . . 8  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  A } ) )
3431, 33rexprg 4047 . . . . . . 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 454 . . . . . 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 1024 . . . . 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 4077 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
3837eqeq2d 2436 . . . . . . 7  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
39 preq2 4077 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4039eqeq2d 2436 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  B } ) )
4138, 40rexprg 4047 . . . . . 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 1025 . . . . 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 1334 . . . 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 1026 . . 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 4092 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
4645a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
4746difeq1d 3582 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
48 necom 2693 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
49 necom 2693 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
50 diftpsn3 4135 . . . . . . . . 9  |-  ( ( B  =/=  A  /\  C  =/=  A )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
5148, 49, 50syl2anb 481 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
52513adant3 1025 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5347, 52eqtrd 2463 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5453rexeqdv 3032 . . . . 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 4092 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
5655eqcomi 2435 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
5756a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
5857difeq1d 3582 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
59 necom 2693 . . . . . . . . . . . 12  |-  ( B  =/=  C  <->  C  =/=  B )
6059anbi1i 699 . . . . . . . . . . 11  |-  ( ( B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  B  /\  A  =/= 
B ) )
6160biimpi 197 . . . . . . . . . 10  |-  ( ( B  =/=  C  /\  A  =/=  B )  -> 
( C  =/=  B  /\  A  =/=  B
) )
6261ancoms 454 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( C  =/=  B  /\  A  =/=  B
) )
63 diftpsn3 4135 . . . . . . . . 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 1024 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
6658, 65eqtrd 2463 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
6766rexeqdv 3032 . . . . 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 4135 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
69683adant1 1023 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7069rexeqdv 3032 . . . . 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 1334 . . . 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 1028 . . 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 4075 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7473eqeq2i 2440 . . . . . . 7  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  B }  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C } )
7574orbi2i 521 . . . . . 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 516 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  \/  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)
7775, 76bitr2i 253 . . . . 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 25149 . . . . . . . . . 10  |-  ( V USGrph  E  ->  A  e/  ( <. V ,  E >. Neighbors  A
) )
80 df-nel 2621 . . . . . . . . . . 11  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  <->  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )
81 prid2g 4104 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
82813ad2ant1 1026 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
83 eleq2 2495 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { B ,  A } ) )
8482, 83syl5ibrcom 225 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { B ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
8584con3rr3 141 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8680, 85sylbi 198 . . . . . . . . . 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 467 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
) )
8988impcom 431 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { B ,  A }
)
90893adant3 1025 . . . . . 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 406 . . . . . . 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 388 . . . . . . 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 264 . . . . . 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 4104 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
96953ad2ant1 1026 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
97 eleq2 2495 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { C ,  A }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { C ,  A } ) )
9896, 97syl5ibrcom 225 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { C ,  A }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
9998con3rr3 141 . . . . . . . . . . 11  |-  ( -.  A  e.  ( <. V ,  E >. Neighbors  A
)  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
10080, 99sylbi 198 . . . . . . . . . 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 467 . . . . . . . 8  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
) )
103102impcom 431 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { C ,  A }
)
1041033adant3 1025 . . . . . 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 406 . . . . . 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 714 . . . 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 4103 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1091083ad2ant1 1026 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
110 eleq2 2495 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  B }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  B } ) )
111109, 110syl5ibrcom 225 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  B }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
112111con3dimp 442 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  B }
)
113 prid1g 4103 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1141133ad2ant1 1026 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
115 eleq2 2495 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { A ,  C }  ->  ( A  e.  ( <. V ,  E >. Neighbors  A )  <-> 
A  e.  { A ,  C } ) )
116114, 115syl5ibrcom 225 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( <. V ,  E >. Neighbors  A )  =  { A ,  C }  ->  A  e.  ( <. V ,  E >. Neighbors  A
) ) )
117116con3dimp 442 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )  ->  -.  ( <. V ,  E >. Neighbors  A )  =  { A ,  C }
)
118112, 117jca 534 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 198 . . . . . . . . . 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 467 . . . . . . . 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 431 . . . . . . 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 1025 . . . . . 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 492 . . . . . 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 215 . . . . 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 1370 . . . 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 282 . . 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 289 . 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 289 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 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618    e/ wnel 2619   E.wrex 2776    \ cdif 3433   {csn 3996   {cpr 3998   {ctp 4000   <.cop 4002   class class class wbr 4420  (class class class)co 6301   USGrph cusg 25043   Neighbors cnbgra 25130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12515  df-usgra 25046  df-nbgra 25133
This theorem is referenced by:  nb3grapr  25166
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