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Theorem nb3graprlem2 23495
Description: Lemma 2 for nb3grapr 23496. (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 3985 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
21difeq2d 3572 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
3 preq1 4052 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
43eqeq2d 2465 . . . . 5  |-  ( v  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  w } ) )
52, 4rexeqbidv 3028 . . . 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 3985 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
76difeq2d 3572 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
8 preq1 4052 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
98eqeq2d 2465 . . . . 5  |-  ( v  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  w } ) )
107, 9rexeqbidv 3028 . . . 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 3985 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1211difeq2d 3572 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
13 preq1 4052 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1413eqeq2d 2465 . . . . 5  |-  ( v  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  {
v ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  w } ) )
1512, 14rexeqbidv 3028 . . . 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 1009 . 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 3565 . . . . . 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 3020 . . . 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 3028 . . 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 1010 . 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 4053 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2524eqeq2d 2465 . . . . . . 7  |-  ( w  =  B  ->  (
( <. V ,  E >. Neighbors  A )  =  { A ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { A ,  B } ) )
26 preq2 4053 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
2726eqeq2d 2465 . . . . . . 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 1006 . . . . 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 4053 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3130eqeq2d 2465 . . . . . . . 8  |-  ( w  =  C  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { B ,  C } ) )
32 preq2 4053 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3332eqeq2d 2465 . . . . . . . 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 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 1007 . . . . 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 4053 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
3837eqeq2d 2465 . . . . . . 7  |-  ( w  =  A  ->  (
( <. V ,  E >. Neighbors  A )  =  { C ,  w }  <->  (
<. V ,  E >. Neighbors  A
)  =  { C ,  A } ) )
39 preq2 4053 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4039eqeq2d 2465 . . . . . . 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 1008 . . . . 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 1289 . . . 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 1009 . . 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 4068 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
4645a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
4746difeq1d 3571 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
48 necom 2717 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
49 necom 2717 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
50 diftpsn3 4110 . . . . . . . . 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 1008 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5347, 52eqtrd 2492 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5453rexeqdv 3020 . . . . 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 4068 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
5655eqcomi 2464 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
5756a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
5857difeq1d 3571 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
59 necom 2717 . . . . . . . . . . . 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 4110 . . . . . . . . 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 1007 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
6658, 65eqtrd 2492 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
6766rexeqdv 3020 . . . . 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 4110 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
69683adant1 1006 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7069rexeqdv 3020 . . . . 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 1289 . . . 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 1011 . . 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 4051 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7473eqeq2i 2469 . . . . . . 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 23482 . . . . . . . . . 10  |-  ( V USGrph  E  ->  A  e/  ( <. V ,  E >. Neighbors  A
) )
80 df-nel 2647 . . . . . . . . . . 11  |-  ( A  e/  ( <. V ,  E >. Neighbors  A )  <->  -.  A  e.  ( <. V ,  E >. Neighbors  A ) )
81 prid2g 4080 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
82813ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
83 eleq2 2524 . . . . . . . . . . . . 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 1008 . . . . . 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 4080 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
96953ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
97 eleq2 2524 . . . . . . . . . . . . 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 1008 . . . . . 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 4079 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1091083ad2ant1 1009 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
110 eleq2 2524 . . . . . . . . . . . . . . 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 4079 . . . . . . . . . . . . . . . 16  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1141133ad2ant1 1009 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
115 eleq2 2524 . . . . . . . . . . . . . . 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 1008 . . . . . 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 1325 . . . 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 964    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    e/ wnel 2645   E.wrex 2796    \ cdif 3423   {csn 3975   {cpr 3977   {ctp 3979   <.cop 3981   class class class wbr 4390  (class class class)co 6190   USGrph cusg 23399   Neighbors cnbgra 23464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-hash 12205  df-usgra 23401  df-nbgra 23467
This theorem is referenced by:  nb3grapr  23496
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