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Theorem nb3grapr 21415
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 20 . . . . . 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 3842 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
32eleq1i 2467 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
4 prcom 3842 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
54eleq1i 2467 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
6 prcom 3842 . . . . . . . . . 10  |-  { C ,  A }  =  { A ,  C }
76eleq1i 2467 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
83, 5, 73anbi123i 1142 . . . . . . . 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 941 . . . . . . . 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 244 . . . . . . 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 624 . . . . 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 1263 . . . . 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 241 . . . 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 21413 . . . . 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 941 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X )
)
1817biimpi 187 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
) )
19 tprot 3859 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
2019eqeq2i 2414 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  C ,  A }
)
2120biimpi 187 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  C ,  A }
)
2221anim1i 552 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  C ,  A }  /\  V USGrph  E
) )
23 nb3graprlem1 21413 . . . . . 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 464 . . . . 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 941 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2625biimpri 198 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
27 tprot 3859 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2827eqcomi 2408 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2928eqeq2i 2414 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
3029biimpi 187 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
3130anim1i 552 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
32 nb3graprlem1 21413 . . . . . 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 464 . . . . 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 1254 . . . 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 977 . . 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 21414 . . . 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 677 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  <->  ( V  =  { B ,  C ,  A }  /\  V USGrph  E ) )
38 necom 2648 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
39 necom 2648 . . . . . . 7  |-  ( A  =/=  C  <->  C  =/=  A )
40 biid 228 . . . . . . 7  |-  ( B  =/=  C  <->  B  =/=  C )
4138, 39, 403anbi123i 1142 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
42 3anrot 941 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  A  /\  C  =/=  A )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
4341, 42bitr4i 244 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  C  /\  B  =/= 
A  /\  C  =/=  A ) )
44 nb3graprlem2 21414 . . . . 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 1227 . . . 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 20 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { A ,  B ,  C }
)
4746, 28syl6eq 2452 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
4847anim1i 552 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
49 3anrot 941 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( A  =/=  C  /\  B  =/= 
C  /\  A  =/=  B ) )
50 necom 2648 . . . . . . . 8  |-  ( B  =/=  C  <->  C  =/=  B )
51 biid 228 . . . . . . . 8  |-  ( A  =/=  B  <->  A  =/=  B )
5239, 50, 513anbi123i 1142 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5349, 52bitri 241 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5453biimpi 187 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( C  =/=  A  /\  C  =/=  B  /\  A  =/= 
B ) )
55 nb3graprlem2 21414 . . . . 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 1226 . . . 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 1254 . . 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 273 . 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 6048 . . . . . 6  |-  ( x  =  A  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  A
) )
6059eqeq1d 2412 . . . . 5  |-  ( x  =  A  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
61602rexbidv 2709 . . . 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 6048 . . . . . 6  |-  ( x  =  B  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  B
) )
6362eqeq1d 2412 . . . . 5  |-  ( x  =  B  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
64632rexbidv 2709 . . . 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 6048 . . . . . 6  |-  ( x  =  C  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  C
) )
6665eqeq1d 2412 . . . . 5  |-  ( x  =  C  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
67662rexbidv 2709 . . . 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 3819 . . 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 978 . 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 2864 . . . . 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 193 . . . 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 452 . . 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 979 . 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 273 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277   {csn 3774   {cpr 3775   {ctp 3776   <.cop 3777   class class class wbr 4172   ran crn 4838  (class class class)co 6040   USGrph cusg 21318   Neighbors cnbgra 21383
This theorem is referenced by:  cusgra3vnbpr  21427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-usgra 21320  df-nbgra 21386
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