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Theorem nb3grapr 24882
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 23 . . . . . 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 4052 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
32eleq1i 2481 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
4 prcom 4052 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
54eleq1i 2481 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
6 prcom 4052 . . . . . . . . . 10  |-  { C ,  A }  =  { A ,  C }
76eleq1i 2481 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
83, 5, 73anbi123i 1188 . . . . . . . 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 981 . . . . . . . 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 254 . . . . . . 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 1312 . . . . 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 251 . . . 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 24880 . . . . 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 981 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X )
)
1817biimpi 196 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
) )
19 tprot 4069 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
2019eqeq2i 2422 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  C ,  A }
)
2120biimpi 196 . . . . . . 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 24880 . . . . . 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 981 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2625biimpri 208 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
27 tprot 4069 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2827eqcomi 2417 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2928eqeq2i 2422 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
3029biimpi 196 . . . . . . 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 24880 . . . . . 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 1303 . . . 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 1019 . . 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 24881 . . . 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 2674 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
39 necom 2674 . . . . . . 7  |-  ( A  =/=  C  <->  C  =/=  A )
40 biid 238 . . . . . . 7  |-  ( B  =/=  C  <->  B  =/=  C )
4138, 39, 403anbi123i 1188 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
42 3anrot 981 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  A  /\  C  =/=  A )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
4341, 42bitr4i 254 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  C  /\  B  =/= 
A  /\  C  =/=  A ) )
44 nb3graprlem2 24881 . . . . 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 1275 . . . 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 23 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { A ,  B ,  C }
)
4746, 28syl6eq 2461 . . . . . 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 981 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( A  =/=  C  /\  B  =/= 
C  /\  A  =/=  B ) )
50 necom 2674 . . . . . . . 8  |-  ( B  =/=  C  <->  C  =/=  B )
51 biid 238 . . . . . . . 8  |-  ( A  =/=  B  <->  A  =/=  B )
5239, 50, 513anbi123i 1188 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5349, 52bitri 251 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5453biimpi 196 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( C  =/=  A  /\  C  =/=  B  /\  A  =/= 
B ) )
55 nb3graprlem2 24881 . . . . 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 1274 . . . 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 1303 . . 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 283 . 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 6288 . . . . . 6  |-  ( x  =  A  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  A
) )
6059eqeq1d 2406 . . . . 5  |-  ( x  =  A  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
61602rexbidv 2927 . . . 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 6288 . . . . . 6  |-  ( x  =  B  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  B
) )
6362eqeq1d 2406 . . . . 5  |-  ( x  =  B  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
64632rexbidv 2927 . . . 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 6288 . . . . . 6  |-  ( x  =  C  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  C
) )
6665eqeq1d 2406 . . . . 5  |-  ( x  =  C  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
67662rexbidv 2927 . . . 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 4025 . . 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 1020 . 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 3006 . . . . 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 203 . . . 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 1021 . 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 283 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   E.wrex 2757    \ cdif 3413   {csn 3974   {cpr 3976   {ctp 3978   <.cop 3980   class class class wbr 4397   ran crn 4826  (class class class)co 6280   USGrph cusg 24759   Neighbors cnbgra 24846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-hash 12455  df-usgra 24762  df-nbgra 24849
This theorem is referenced by:  cusgra3vnbpr  24894
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