MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nb3grapr Structured version   Unicode version

Theorem nb3grapr 25173
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 4076 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
32eleq1i 2500 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
4 prcom 4076 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
54eleq1i 2500 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
6 prcom 4076 . . . . . . . . . 10  |-  { C ,  A }  =  { A ,  C }
76eleq1i 2500 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
83, 5, 73anbi123i 1195 . . . . . . . 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 988 . . . . . . . 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 256 . . . . . . 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 647 . . . . 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 1345 . . . . 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 253 . . . 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 25171 . . . . 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 988 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X )
)
1817biimpi 198 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  C  e.  Z  /\  A  e.  X
) )
19 tprot 4093 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
2019eqeq2i 2441 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  C ,  A }
)
2120biimpi 198 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  C ,  A }
)
2221anim1i 571 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  C ,  A }  /\  V USGrph  E
) )
23 nb3graprlem1 25171 . . . . . 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 480 . . . . 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 988 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2625biimpri 210 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
27 tprot 4093 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2827eqcomi 2436 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2928eqeq2i 2441 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
3029biimpi 198 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
3130anim1i 571 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
32 nb3graprlem1 25171 . . . . . 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 480 . . . . 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 1336 . . . 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 1026 . . 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 25172 . . . 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 700 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  <->  ( V  =  { B ,  C ,  A }  /\  V USGrph  E ) )
38 necom 2694 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
39 necom 2694 . . . . . . 7  |-  ( A  =/=  C  <->  C  =/=  A )
40 biid 240 . . . . . . 7  |-  ( B  =/=  C  <->  B  =/=  C )
4138, 39, 403anbi123i 1195 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
42 3anrot 988 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  A  /\  C  =/=  A )  <->  ( B  =/=  A  /\  C  =/= 
A  /\  B  =/=  C ) )
4341, 42bitr4i 256 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( B  =/=  C  /\  B  =/= 
A  /\  C  =/=  A ) )
44 nb3graprlem2 25172 . . . . 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 1308 . . . 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 2480 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
4847anim1i 571 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
49 3anrot 988 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( A  =/=  C  /\  B  =/= 
C  /\  A  =/=  B ) )
50 necom 2694 . . . . . . . 8  |-  ( B  =/=  C  <->  C  =/=  B )
51 biid 240 . . . . . . . 8  |-  ( A  =/=  B  <->  A  =/=  B )
5239, 50, 513anbi123i 1195 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5349, 52bitri 253 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( C  =/=  A  /\  C  =/= 
B  /\  A  =/=  B ) )
5453biimpi 198 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( C  =/=  A  /\  C  =/=  B  /\  A  =/= 
B ) )
55 nb3graprlem2 25172 . . . . 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 1307 . . . 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 1336 . . 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 285 . 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 6311 . . . . . 6  |-  ( x  =  A  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  A
) )
6059eqeq1d 2425 . . . . 5  |-  ( x  =  A  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  A )  =  {
y ,  z } ) )
61602rexbidv 2947 . . . 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 6311 . . . . . 6  |-  ( x  =  B  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  B
) )
6362eqeq1d 2425 . . . . 5  |-  ( x  =  B  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  B )  =  {
y ,  z } ) )
64632rexbidv 2947 . . . 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 6311 . . . . . 6  |-  ( x  =  C  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  C
) )
6665eqeq1d 2425 . . . . 5  |-  ( x  =  C  ->  (
( <. V ,  E >. Neighbors  x )  =  {
y ,  z }  <-> 
( <. V ,  E >. Neighbors  C )  =  {
y ,  z } ) )
67662rexbidv 2947 . . . 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 4049 . . 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 1027 . 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 3026 . . . . 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 205 . . . 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 467 . . 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 1028 . 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 285 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   E.wrex 2777    \ cdif 3434   {csn 3997   {cpr 3999   {ctp 4001   <.cop 4003   class class class wbr 4421   ran crn 4852  (class class class)co 6303   USGrph cusg 25049   Neighbors cnbgra 25137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-hash 12517  df-usgra 25052  df-nbgra 25140
This theorem is referenced by:  cusgra3vnbpr  25185
  Copyright terms: Public domain W3C validator