Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nb3grprlem2 Structured version   Visualization version   Unicode version

Theorem nb3grprlem2 39619
Description: Lemma 2 for nb3grapr 25260. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v  |-  V  =  (Vtx `  G )
nb3grpr.e  |-  E  =  (Edg `  G )
nb3grpr.g  |-  ( ph  ->  G  e. USGraph  )
nb3grpr.t  |-  ( ph  ->  V  =  { A ,  B ,  C }
)
nb3grpr.s  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
nb3grpr.n  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
Assertion
Ref Expression
nb3grprlem2  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } ) )
Distinct variable groups:    v, A    v, B    v, C    v, E    v, G    v, V    ph, v    w, A, v   
w, B    w, C    w, G    w, V
Allowed substitution hints:    ph( w)    E( w)    X( w, v)    Y( w, v)    Z( w, v)

Proof of Theorem nb3grprlem2
StepHypRef Expression
1 nb3grpr.s . . 3  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
2 sneq 3969 . . . . . 6  |-  ( v  =  A  ->  { v }  =  { A } )
32difeq2d 3540 . . . . 5  |-  ( v  =  A  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { A } ) )
4 preq1 4042 . . . . . 6  |-  ( v  =  A  ->  { v ,  w }  =  { A ,  w }
)
54eqeq2d 2481 . . . . 5  |-  ( v  =  A  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  w } ) )
63, 5rexeqbidv 2988 . . . 4  |-  ( v  =  A  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }
) )
7 sneq 3969 . . . . . 6  |-  ( v  =  B  ->  { v }  =  { B } )
87difeq2d 3540 . . . . 5  |-  ( v  =  B  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { B } ) )
9 preq1 4042 . . . . . 6  |-  ( v  =  B  ->  { v ,  w }  =  { B ,  w }
)
109eqeq2d 2481 . . . . 5  |-  ( v  =  B  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  w } ) )
118, 10rexeqbidv 2988 . . . 4  |-  ( v  =  B  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }
) )
12 sneq 3969 . . . . . 6  |-  ( v  =  C  ->  { v }  =  { C } )
1312difeq2d 3540 . . . . 5  |-  ( v  =  C  ->  ( { A ,  B ,  C }  \  { v } )  =  ( { A ,  B ,  C }  \  { C } ) )
14 preq1 4042 . . . . . 6  |-  ( v  =  C  ->  { v ,  w }  =  { C ,  w }
)
1514eqeq2d 2481 . . . . 5  |-  ( v  =  C  ->  (
( G NeighbVtx  A )  =  { v ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  w } ) )
1613, 15rexeqbidv 2988 . . . 4  |-  ( v  =  C  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w }
) )
176, 11, 16rextpg 4015 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } 
<->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
181, 17syl 17 . 2  |-  ( ph  ->  ( E. v  e. 
{ A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } 
<->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
19 nb3grpr.t . . . 4  |-  ( ph  ->  V  =  { A ,  B ,  C }
)
20 nb3grpr.g . . . 4  |-  ( ph  ->  G  e. USGraph  )
2119, 20jca 541 . . 3  |-  ( ph  ->  ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )
22 simpl 464 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  ->  V  =  { A ,  B ,  C }
)
23 difeq1 3533 . . . . . 6  |-  ( V  =  { A ,  B ,  C }  ->  ( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2423adantr 472 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( V  \  {
v } )  =  ( { A ,  B ,  C }  \  { v } ) )
2524rexeqdv 2980 . . . 4  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
2622, 25rexeqbidv 2988 . . 3  |-  ( ( V  =  { A ,  B ,  C }  /\  G  e. USGraph  )  -> 
( E. v  e.  V  E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
2721, 26syl 17 . 2  |-  ( ph  ->  ( E. v  e.  V  E. w  e.  ( V  \  {
v } ) ( G NeighbVtx  A )  =  {
v ,  w }  <->  E. v  e.  { A ,  B ,  C } E. w  e.  ( { A ,  B ,  C }  \  { v } ) ( G NeighbVtx  A )  =  {
v ,  w }
) )
28 preq2 4043 . . . . . . . 8  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
2928eqeq2d 2481 . . . . . . 7  |-  ( w  =  B  ->  (
( G NeighbVtx  A )  =  { A ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  B } ) )
30 preq2 4043 . . . . . . . 8  |-  ( w  =  C  ->  { A ,  w }  =  { A ,  C }
)
3130eqeq2d 2481 . . . . . . 7  |-  ( w  =  C  ->  (
( G NeighbVtx  A )  =  { A ,  w } 
<->  ( G NeighbVtx  A )  =  { A ,  C } ) )
3229, 31rexprg 4013 . . . . . 6  |-  ( ( B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  <->  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) ) )
33323adant1 1048 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  <->  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) ) )
34 preq2 4043 . . . . . . . . 9  |-  ( w  =  C  ->  { B ,  w }  =  { B ,  C }
)
3534eqeq2d 2481 . . . . . . . 8  |-  ( w  =  C  ->  (
( G NeighbVtx  A )  =  { B ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  C } ) )
36 preq2 4043 . . . . . . . . 9  |-  ( w  =  A  ->  { B ,  w }  =  { B ,  A }
)
3736eqeq2d 2481 . . . . . . . 8  |-  ( w  =  A  ->  (
( G NeighbVtx  A )  =  { B ,  w } 
<->  ( G NeighbVtx  A )  =  { B ,  A } ) )
3835, 37rexprg 4013 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
3938ancoms 460 . . . . . 6  |-  ( ( A  e.  X  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
40393adant2 1049 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
41 preq2 4043 . . . . . . . 8  |-  ( w  =  A  ->  { C ,  w }  =  { C ,  A }
)
4241eqeq2d 2481 . . . . . . 7  |-  ( w  =  A  ->  (
( G NeighbVtx  A )  =  { C ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  A } ) )
43 preq2 4043 . . . . . . . 8  |-  ( w  =  B  ->  { C ,  w }  =  { C ,  B }
)
4443eqeq2d 2481 . . . . . . 7  |-  ( w  =  B  ->  (
( G NeighbVtx  A )  =  { C ,  w } 
<->  ( G NeighbVtx  A )  =  { C ,  B } ) )
4542, 44rexprg 4013 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
46453adant3 1050 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. w  e. 
{ A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
4733, 40, 463orbi123d 1364 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( (
( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A }
)  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) ) )
481, 47syl 17 . . 3  |-  ( ph  ->  ( ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( (
( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A }
)  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) ) )
49 nb3grpr.n . . . 4  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
50 tprot 4058 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
5150a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { B ,  C ,  A } )
5251difeq1d 3539 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  ( { B ,  C ,  A }  \  { A } ) )
53 necom 2696 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
54 necom 2696 . . . . . . . . 9  |-  ( A  =/=  C  <->  C  =/=  A )
55 diftpsn3 4101 . . . . . . . . 9  |-  ( ( B  =/=  A  /\  C  =/=  A )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
5653, 54, 55syl2anb 487 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C )  -> 
( { B ,  C ,  A }  \  { A } )  =  { B ,  C } )
57563adant3 1050 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { B ,  C ,  A }  \  { A } )  =  { B ,  C }
)
5852, 57eqtrd 2505 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { A } )  =  { B ,  C }
)
5958rexeqdv 2980 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  <->  E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }
) )
60 tprot 4058 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
6160eqcomi 2480 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
6261a1i 11 . . . . . . . 8  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  { C ,  A ,  B } )
6362difeq1d 3539 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  ( { C ,  A ,  B }  \  { B } ) )
64 necom 2696 . . . . . . . . . . . 12  |-  ( B  =/=  C  <->  C  =/=  B )
6564anbi1i 709 . . . . . . . . . . 11  |-  ( ( B  =/=  C  /\  A  =/=  B )  <->  ( C  =/=  B  /\  A  =/= 
B ) )
6665biimpi 199 . . . . . . . . . 10  |-  ( ( B  =/=  C  /\  A  =/=  B )  -> 
( C  =/=  B  /\  A  =/=  B
) )
6766ancoms 460 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( C  =/=  B  /\  A  =/=  B
) )
68 diftpsn3 4101 . . . . . . . . 9  |-  ( ( C  =/=  B  /\  A  =/=  B )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
6967, 68syl 17 . . . . . . . 8  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { C ,  A ,  B }  \  { B } )  =  { C ,  A } )
70693adant2 1049 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { C ,  A ,  B }  \  { B } )  =  { C ,  A }
)
7163, 70eqtrd 2505 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { B } )  =  { C ,  A }
)
7271rexeqdv 2980 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  <->  E. w  e.  { C ,  A }  ( G NeighbVtx  A )  =  { B ,  w }
) )
73 diftpsn3 4101 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
74733adant1 1048 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B }
)
7574rexeqdv 2980 . . . . 5  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w }  <->  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w }
) )
7659, 72, 753orbi123d 1364 . . . 4  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  (
( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } ) ) )
7749, 76syl 17 . . 3  |-  ( ph  ->  ( ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } )  <->  ( E. w  e.  { B ,  C }  ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  { C ,  A } 
( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  { A ,  B }  ( G NeighbVtx  A )  =  { C ,  w } ) ) )
78 prcom 4041 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
7978eqeq2i 2483 . . . . . . 7  |-  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( G NeighbVtx  A )  =  { B ,  C }
)
8079orbi2i 528 . . . . . 6  |-  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  C } ) )
81 oridm 523 . . . . . 6  |-  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  C } )  <->  ( G NeighbVtx  A )  =  { B ,  C } )
8280, 81bitr2i 258 . . . . 5  |-  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) )
8382a1i 11 . . . 4  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
84 usgrnbnself2 39598 . . . . . . . 8  |-  ( G  e. USGraph  ->  A  e/  ( G NeighbVtx  A ) )
85 df-nel 2644 . . . . . . . . 9  |-  ( A  e/  ( G NeighbVtx  A )  <->  -.  A  e.  ( G NeighbVtx  A ) )
86 prid2g 4070 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  { B ,  A } )
87863ad2ant1 1051 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { B ,  A } )
88 eleq2 2538 . . . . . . . . . . 11  |-  ( ( G NeighbVtx  A )  =  { B ,  A }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { B ,  A }
) )
8987, 88syl5ibrcom 230 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { B ,  A }  ->  A  e.  ( G NeighbVtx  A )
) )
9089con3rr3 143 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( G NeighbVtx  A )  =  { B ,  A }
) )
9185, 90sylbi 200 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } ) )
9284, 91syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } ) )
9320, 1, 92sylc 61 . . . . . 6  |-  ( ph  ->  -.  ( G NeighbVtx  A )  =  { B ,  A } )
94 biorf 412 . . . . . . 7  |-  ( -.  ( G NeighbVtx  A )  =  { B ,  A }  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  A }  \/  ( G NeighbVtx  A )  =  { B ,  C } ) ) )
95 orcom 394 . . . . . . 7  |-  ( ( ( G NeighbVtx  A )  =  { B ,  A }  \/  ( G NeighbVtx  A )  =  { B ,  C } )  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) )
9694, 95syl6bb 269 . . . . . 6  |-  ( -.  ( G NeighbVtx  A )  =  { B ,  A }  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
9793, 96syl 17 . . . . 5  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } ) ) )
98 prid2g 4070 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  { C ,  A } )
99983ad2ant1 1051 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { C ,  A } )
100 eleq2 2538 . . . . . . . . . . 11  |-  ( ( G NeighbVtx  A )  =  { C ,  A }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { C ,  A }
) )
10199, 100syl5ibrcom 230 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { C ,  A }  ->  A  e.  ( G NeighbVtx  A )
) )
102101con3rr3 143 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  -.  ( G NeighbVtx  A )  =  { C ,  A }
) )
10385, 102sylbi 200 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } ) )
10484, 103syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } ) )
10520, 1, 104sylc 61 . . . . . 6  |-  ( ph  ->  -.  ( G NeighbVtx  A )  =  { C ,  A } )
106 biorf 412 . . . . . 6  |-  ( -.  ( G NeighbVtx  A )  =  { C ,  A }  ->  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
107105, 106syl 17 . . . . 5  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { C ,  B }  <->  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) ) )
10897, 107orbi12d 724 . . . 4  |-  ( ph  ->  ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { C ,  B } )  <->  ( (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
109 prid1g 4069 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { A ,  B } )
1101093ad2ant1 1051 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  B } )
111 eleq2 2538 . . . . . . . . . . . . 13  |-  ( ( G NeighbVtx  A )  =  { A ,  B }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { A ,  B }
) )
112110, 111syl5ibrcom 230 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { A ,  B }  ->  A  e.  ( G NeighbVtx  A )
) )
113112con3dimp 448 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  -.  ( G NeighbVtx  A )  =  { A ,  B }
)
114 prid1g 4069 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  A  e.  { A ,  C } )
1151143ad2ant1 1051 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  A  e.  { A ,  C } )
116 eleq2 2538 . . . . . . . . . . . . 13  |-  ( ( G NeighbVtx  A )  =  { A ,  C }  ->  ( A  e.  ( G NeighbVtx  A )  <->  A  e.  { A ,  C }
) )
117115, 116syl5ibrcom 230 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( G NeighbVtx  A )  =  { A ,  C }  ->  A  e.  ( G NeighbVtx  A )
) )
118117con3dimp 448 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  -.  ( G NeighbVtx  A )  =  { A ,  C }
)
119113, 118jca 541 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  -.  A  e.  ( G NeighbVtx  A )
)  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) )
120119expcom 442 . . . . . . . . 9  |-  ( -.  A  e.  ( G NeighbVtx  A )  ->  (
( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12185, 120sylbi 200 . . . . . . . 8  |-  ( A  e/  ( G NeighbVtx  A )  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12284, 121syl 17 . . . . . . 7  |-  ( G  e. USGraph  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C }
) ) )
12320, 1, 122sylc 61 . . . . . 6  |-  ( ph  ->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C } ) )
124 ioran 498 . . . . . 6  |-  ( -.  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C }
)  <->  ( -.  ( G NeighbVtx  A )  =  { A ,  B }  /\  -.  ( G NeighbVtx  A )  =  { A ,  C } ) )
125123, 124sylibr 217 . . . . 5  |-  ( ph  ->  -.  ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } ) )
1261253bior1fd 1403 . . . 4  |-  ( ph  ->  ( ( ( ( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  (
( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B } ) )  <-> 
( ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
12783, 108, 1263bitrd 287 . . 3  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( ( ( G NeighbVtx  A )  =  { A ,  B }  \/  ( G NeighbVtx  A )  =  { A ,  C } )  \/  (
( G NeighbVtx  A )  =  { B ,  C }  \/  ( G NeighbVtx  A )  =  { B ,  A } )  \/  ( ( G NeighbVtx  A )  =  { C ,  A }  \/  ( G NeighbVtx  A )  =  { C ,  B }
) ) ) )
12848, 77, 1273bitr4rd 294 . 2  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  ( E. w  e.  ( { A ,  B ,  C }  \  { A } ) ( G NeighbVtx  A )  =  { A ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { B } ) ( G NeighbVtx  A )  =  { B ,  w }  \/  E. w  e.  ( { A ,  B ,  C }  \  { C } ) ( G NeighbVtx  A )  =  { C ,  w } ) ) )
12918, 27, 1283bitr4rd 294 1  |-  ( ph  ->  ( ( G NeighbVtx  A )  =  { B ,  C }  <->  E. v  e.  V  E. w  e.  ( V  \  { v } ) ( G NeighbVtx  A )  =  { v ,  w } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    e/ wnel 2642   E.wrex 2757    \ cdif 3387   {csn 3959   {cpr 3961   {ctp 3963   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251  Edgcedga 39371   USGraph cusgr 39397   NeighbVtx cnbgr 39561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-nbgr 39565
This theorem is referenced by:  nb3grpr  39620
  Copyright terms: Public domain W3C validator