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Theorem nb3grapr 23360
Description: The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
Assertion
Ref Expression
nb3grapr  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  x )  =  {
y ,  z } ) )
Distinct variable groups:    x, A, y, z    x, B, y, z    x, C, y, z    x, E, y, z    x, V, y, z
Allowed substitution hints:    X( x, y, z)    Y( x, y, z)    Z( x, y, z)

Proof of Theorem nb3grapr
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )
2 prcom 3952 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
32eleq1i 2505 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
4 prcom 3952 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
54eleq1i 2505 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
6 prcom 3952 . . . . . . . . . 10  |-  { C ,  A }  =  { A ,  C }
76eleq1i 2505 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
83, 5, 73anbi123i 1176 . . . . . . . 8  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
9 3anrot 970 . . . . . . . 8  |-  ( ( { A ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  <->  ( { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
108, 9bitr4i 252 . . . . . . 7  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( { A ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
1110a1i 11 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( { A ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
121, 11biadan2 642 . . . . 5  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
13 an6 1298 . . . . 5  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  <-> 
( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
1412, 13bitri 249 . . . 4  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
1514a1i 11 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
16 nb3graprlem1 23358 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
17 3anrot 970 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X )
)
1817biimpi 194 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
) )
19 tprot 3969 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
2019eqeq2i 2452 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  C ,  A }
)
2120biimpi 194 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  C ,  A }
)
2221anim1i 568 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  C ,  A }  /\  V USGrph  E
) )
23 nb3graprlem1 23358 . . . . . 6  |-  ( ( ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
)  /\  ( V  =  { B ,  C ,  A }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  B )  =  { C ,  A }  <->  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E ) ) )
2418, 22, 23syl2an 477 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  B )  =  { C ,  A }  <->  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E ) ) )
25 3anrot 970 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2625biimpri 206 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
27 tprot 3969 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2827eqcomi 2446 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2928eqeq2i 2452 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
3029biimpi 194 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
3130anim1i 568 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
32 nb3graprlem1 23358 . . . . . 6  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( V  =  { C ,  A ,  B }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  C )  =  { A ,  B }  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
3326, 31, 32syl2an 477 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  C )  =  { A ,  B }  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
3416, 24, 333anbi123d 1289 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { C ,  A }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
)  <->  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
35343adant3 1008 . . 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  B
)  =  { C ,  A }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } )  <->  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  C }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
36 nb3graprlem2 23359 . . . 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 }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
3720anbi1i 695 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  <->  ( V  =  { B ,  C ,  A }  /\  V USGrph  E ) )
38 necom 2692 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
39 necom 2692 . . . . . . 7  |-  ( A  =/=  C  <->  C  =/=  A )
40 biid 236 . . . . . . 7  |-  ( B  =/=  C  <->  B  =/=  C )
4138, 39, 403anbi123i 1176 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
42 3anrot 970 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  A  /\  C  =/=  A )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
4341, 42bitr4i 252 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  C  /\  B  =/= 
A  /\  C  =/=  A ) )
44 nb3graprlem2 23359 . . . . 5  |-  ( ( ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
)  /\  ( V  =  { B ,  C ,  A }  /\  V USGrph  E )  /\  ( B  =/=  C  /\  B  =/=  A  /\  C  =/= 
A ) )  -> 
( ( <. V ,  E >. Neighbors  B )  =  { C ,  A }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
4517, 37, 43, 44syl3anb 1261 . . . 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  B )  =  { C ,  A }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
46 id 22 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { A ,  B ,  C }
)
4746, 28syl6eq 2490 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
4847anim1i 568 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
49 3anrot 970 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( A  =/=  C  /\  B  =/= 
C  /\  A  =/=  B ) )
50 necom 2692 . . . . . . . 8  |-  ( B  =/=  C  <->  C  =/=  B )
51 biid 236 . . . . . . . 8  |-  ( A  =/=  B  <->  A  =/=  B )
5239, 50, 513anbi123i 1176 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5349, 52bitri 249 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5453biimpi 194 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( C  =/=  A  /\  C  =/=  B  /\  A  =/= 
B ) )
55 nb3graprlem2 23359 . . . . 5  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( V  =  { C ,  A ,  B }  /\  V USGrph  E )  /\  ( C  =/=  A  /\  C  =/=  B  /\  A  =/= 
B ) )  -> 
( ( <. V ,  E >. Neighbors  C )  =  { A ,  B }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
5626, 48, 54, 55syl3an 1260 . . . 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  C )  =  { A ,  B }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
5736, 45, 563anbi123d 1289 . . 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  B
)  =  { C ,  A }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } )  <->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  A )  =  {
y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  B
)  =  { y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) ) )
5815, 35, 573bitr2d 281 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  A
)  =  { y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  B )  =  {
y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  C
)  =  { y ,  z } ) ) )
59 oveq2 6098 . . . . . 6  |-  ( x  =  A  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  A
) )
6059eqeq1d 2450 . . . . 5  |-  ( x  =  A  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
61602rexbidv 2757 . . . 4  |-  ( x  =  A  ->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
62 oveq2 6098 . . . . . 6  |-  ( x  =  B  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  B
) )
6362eqeq1d 2450 . . . . 5  |-  ( x  =  B  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
64632rexbidv 2757 . . . 4  |-  ( x  =  B  ->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
65 oveq2 6098 . . . . . 6  |-  ( x  =  C  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  C
) )
6665eqeq1d 2450 . . . . 5  |-  ( x  =  C  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
67662rexbidv 2757 . . . 4  |-  ( x  =  C  ->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <->  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
6861, 64, 67raltpg 3926 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( A. x  e. 
{ A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  A )  =  {
y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  B
)  =  { y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) ) )
69683ad2ant1 1009 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( A. x  e. 
{ A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  ( E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  A )  =  {
y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  B
)  =  { y ,  z }  /\  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) ) )
70 raleq 2916 . . . . 5  |-  ( V  =  { A ,  B ,  C }  ->  ( A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  A. x  e.  { A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z } ) )
7170bicomd 201 . . . 4  |-  ( V  =  { A ,  B ,  C }  ->  ( A. x  e. 
{ A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z } ) )
7271adantr 465 . . 3  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( A. x  e. 
{ A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z } ) )
73723ad2ant2 1010 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( A. x  e. 
{ A ,  B ,  C } E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z }  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  {
y } ) (
<. V ,  E >. Neighbors  x
)  =  { y ,  z } ) )
7458, 69, 733bitr2d 281 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  A. x  e.  V  E. y  e.  V  E. z  e.  ( V  \  { y } ) ( <. V ,  E >. Neighbors  x )  =  {
y ,  z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   E.wrex 2715    \ cdif 3324   {csn 3876   {cpr 3878   {ctp 3880   <.cop 3882   class class class wbr 4291   ran crn 4840  (class class class)co 6090   USGrph cusg 23263   Neighbors cnbgra 23328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-hash 12103  df-usgra 23265  df-nbgra 23331
This theorem is referenced by:  cusgra3vnbpr  23372
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