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Theorem nbuhgr2vtx1edgb 39584
Description: If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)
Hypotheses
Ref Expression
nbgr2vtx1edg.v  |-  V  =  (Vtx `  G )
nbgr2vtx1edg.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
nbuhgr2vtx1edgb  |-  ( ( G  e. UHGraph  /\  ( # `
 V )  =  2 )  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) )
Distinct variable groups:    n, E    n, G, v    n, V, v
Allowed substitution hint:    E( v)

Proof of Theorem nbuhgr2vtx1edgb
Dummy variables  a 
b  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5  |-  V  =  (Vtx `  G )
2 fvex 5889 . . . . 5  |-  (Vtx `  G )  e.  _V
31, 2eqeltri 2545 . . . 4  |-  V  e. 
_V
4 hash2prb 12674 . . . 4  |-  ( V  e.  _V  ->  (
( # `  V )  =  2  <->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  V  =  { a ,  b } ) ) )
53, 4ax-mp 5 . . 3  |-  ( (
# `  V )  =  2  <->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  V  =  { a ,  b } ) )
6 simpr 468 . . . . . . . . . . . 12  |-  ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( a  e.  V  /\  b  e.  V ) )
76ancomd 458 . . . . . . . . . . 11  |-  ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( b  e.  V  /\  a  e.  V ) )
87ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  (
b  e.  V  /\  a  e.  V )
)
9 id 22 . . . . . . . . . . . . 13  |-  ( a  =/=  b  ->  a  =/=  b )
109necomd 2698 . . . . . . . . . . . 12  |-  ( a  =/=  b  ->  b  =/=  a )
1110adantr 472 . . . . . . . . . . 11  |-  ( ( a  =/=  b  /\  V  =  { a ,  b } )  ->  b  =/=  a
)
1211ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  b  =/=  a )
13 prcom 4041 . . . . . . . . . . . . . 14  |-  { a ,  b }  =  { b ,  a }
1413eleq1i 2540 . . . . . . . . . . . . 13  |-  ( { a ,  b }  e.  E  <->  { b ,  a }  e.  E )
1514biimpi 199 . . . . . . . . . . . 12  |-  ( { a ,  b }  e.  E  ->  { b ,  a }  e.  E )
16 sseq2 3440 . . . . . . . . . . . . 13  |-  ( e  =  { b ,  a }  ->  ( { a ,  b }  C_  e  <->  { a ,  b }  C_  { b ,  a } ) )
1716adantl 473 . . . . . . . . . . . 12  |-  ( ( { a ,  b }  e.  E  /\  e  =  { b ,  a } )  ->  ( { a ,  b }  C_  e 
<->  { a ,  b }  C_  { b ,  a } ) )
1813eqimssi 3472 . . . . . . . . . . . . 13  |-  { a ,  b }  C_  { b ,  a }
1918a1i 11 . . . . . . . . . . . 12  |-  ( { a ,  b }  e.  E  ->  { a ,  b }  C_  { b ,  a } )
2015, 17, 19rspcedvd 3143 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  E  ->  E. e  e.  E  { a ,  b }  C_  e )
2120adantl 473 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  E. e  e.  E  { a ,  b }  C_  e )
22 nbgr2vtx1edg.e . . . . . . . . . . . 12  |-  E  =  (Edg `  G )
231, 22nbgrel 39574 . . . . . . . . . . 11  |-  ( G  e. UHGraph  ->  ( b  e.  ( G NeighbVtx  a )  <->  ( ( b  e.  V  /\  a  e.  V
)  /\  b  =/=  a  /\  E. e  e.  E  { a ,  b }  C_  e
) ) )
2423ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  (
b  e.  ( G NeighbVtx  a )  <->  ( (
b  e.  V  /\  a  e.  V )  /\  b  =/=  a  /\  E. e  e.  E  { a ,  b }  C_  e )
) )
258, 12, 21, 24mpbir3and 1213 . . . . . . . . 9  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  b  e.  ( G NeighbVtx  a )
)
266ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  (
a  e.  V  /\  b  e.  V )
)
27 simplrl 778 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  a  =/=  b )
28 id 22 . . . . . . . . . . . 12  |-  ( { a ,  b }  e.  E  ->  { a ,  b }  e.  E )
29 sseq2 3440 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { b ,  a }  C_  e  <->  { b ,  a }  C_  { a ,  b } ) )
3029adantl 473 . . . . . . . . . . . 12  |-  ( ( { a ,  b }  e.  E  /\  e  =  { a ,  b } )  ->  ( { b ,  a }  C_  e 
<->  { b ,  a }  C_  { a ,  b } ) )
31 prcom 4041 . . . . . . . . . . . . . 14  |-  { b ,  a }  =  { a ,  b }
3231eqimssi 3472 . . . . . . . . . . . . 13  |-  { b ,  a }  C_  { a ,  b }
3332a1i 11 . . . . . . . . . . . 12  |-  ( { a ,  b }  e.  E  ->  { b ,  a }  C_  { a ,  b } )
3428, 30, 33rspcedvd 3143 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  E  ->  E. e  e.  E  { b ,  a }  C_  e )
3534adantl 473 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  E. e  e.  E  { b ,  a }  C_  e )
361, 22nbgrel 39574 . . . . . . . . . . 11  |-  ( G  e. UHGraph  ->  ( a  e.  ( G NeighbVtx  b )  <->  ( ( a  e.  V  /\  b  e.  V
)  /\  a  =/=  b  /\  E. e  e.  E  { b ,  a }  C_  e
) ) )
3736ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  (
a  e.  ( G NeighbVtx  b )  <->  ( (
a  e.  V  /\  b  e.  V )  /\  a  =/=  b  /\  E. e  e.  E  { b ,  a }  C_  e )
) )
3826, 27, 35, 37mpbir3and 1213 . . . . . . . . 9  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  a  e.  ( G NeighbVtx  b )
)
3925, 38jca 541 . . . . . . . 8  |-  ( ( ( ( G  e. UHGraph  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  V  =  { a ,  b } ) )  /\  { a ,  b }  e.  E )  ->  (
b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b )
) )
4039ex 441 . . . . . . 7  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( { a ,  b }  e.  E  ->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b )
) ) )
411, 22nbuhgr2vtx1edgblem 39583 . . . . . . . . . . . 12  |-  ( ( G  e. UHGraph  /\  V  =  { a ,  b }  /\  a  e.  ( G NeighbVtx  b )
)  ->  { a ,  b }  e.  E )
42413exp 1230 . . . . . . . . . . 11  |-  ( G  e. UHGraph  ->  ( V  =  { a ,  b }  ->  ( a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) ) )
4342adantr 472 . . . . . . . . . 10  |-  ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( V  =  { a ,  b }  ->  ( a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) ) )
4443adantld 474 . . . . . . . . 9  |-  ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( (
a  =/=  b  /\  V  =  { a ,  b } )  ->  ( a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) ) )
4544imp 436 . . . . . . . 8  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( a  e.  ( G NeighbVtx  b )  ->  { a ,  b }  e.  E ) )
4645adantld 474 . . . . . . 7  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b ) )  ->  { a ,  b }  e.  E ) )
4740, 46impbid 195 . . . . . 6  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( { a ,  b }  e.  E  <->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b )
) ) )
48 eleq1 2537 . . . . . . . . 9  |-  ( V  =  { a ,  b }  ->  ( V  e.  E  <->  { a ,  b }  e.  E ) )
4948adantl 473 . . . . . . . 8  |-  ( ( a  =/=  b  /\  V  =  { a ,  b } )  ->  ( V  e.  E  <->  { a ,  b }  e.  E ) )
50 id 22 . . . . . . . . . 10  |-  ( V  =  { a ,  b }  ->  V  =  { a ,  b } )
51 difeq1 3533 . . . . . . . . . . 11  |-  ( V  =  { a ,  b }  ->  ( V  \  { v } )  =  ( { a ,  b } 
\  { v } ) )
5251raleqdv 2979 . . . . . . . . . 10  |-  ( V  =  { a ,  b }  ->  ( A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v )  <->  A. n  e.  ( { a ,  b }  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) )
5350, 52raleqbidv 2987 . . . . . . . . 9  |-  ( V  =  { a ,  b }  ->  ( A. v  e.  V  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v )  <->  A. v  e.  { a ,  b } A. n  e.  ( { a ,  b }  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) )
54 vex 3034 . . . . . . . . . . 11  |-  a  e. 
_V
55 vex 3034 . . . . . . . . . . 11  |-  b  e. 
_V
56 sneq 3969 . . . . . . . . . . . . 13  |-  ( v  =  a  ->  { v }  =  { a } )
5756difeq2d 3540 . . . . . . . . . . . 12  |-  ( v  =  a  ->  ( { a ,  b }  \  { v } )  =  ( { a ,  b }  \  { a } ) )
58 oveq2 6316 . . . . . . . . . . . . 13  |-  ( v  =  a  ->  ( G NeighbVtx  v )  =  ( G NeighbVtx  a ) )
5958eleq2d 2534 . . . . . . . . . . . 12  |-  ( v  =  a  ->  (
n  e.  ( G NeighbVtx  v )  <->  n  e.  ( G NeighbVtx  a ) ) )
6057, 59raleqbidv 2987 . . . . . . . . . . 11  |-  ( v  =  a  ->  ( A. n  e.  ( { a ,  b }  \  { v } ) n  e.  ( G NeighbVtx  v )  <->  A. n  e.  ( { a ,  b } 
\  { a } ) n  e.  ( G NeighbVtx  a ) ) )
61 sneq 3969 . . . . . . . . . . . . 13  |-  ( v  =  b  ->  { v }  =  { b } )
6261difeq2d 3540 . . . . . . . . . . . 12  |-  ( v  =  b  ->  ( { a ,  b }  \  { v } )  =  ( { a ,  b }  \  { b } ) )
63 oveq2 6316 . . . . . . . . . . . . 13  |-  ( v  =  b  ->  ( G NeighbVtx  v )  =  ( G NeighbVtx  b ) )
6463eleq2d 2534 . . . . . . . . . . . 12  |-  ( v  =  b  ->  (
n  e.  ( G NeighbVtx  v )  <->  n  e.  ( G NeighbVtx  b ) ) )
6562, 64raleqbidv 2987 . . . . . . . . . . 11  |-  ( v  =  b  ->  ( A. n  e.  ( { a ,  b }  \  { v } ) n  e.  ( G NeighbVtx  v )  <->  A. n  e.  ( { a ,  b } 
\  { b } ) n  e.  ( G NeighbVtx  b ) ) )
6654, 55, 60, 65ralpr 4016 . . . . . . . . . 10  |-  ( A. v  e.  { a ,  b } A. n  e.  ( {
a ,  b } 
\  { v } ) n  e.  ( G NeighbVtx  v )  <->  ( A. n  e.  ( {
a ,  b } 
\  { a } ) n  e.  ( G NeighbVtx  a )  /\  A. n  e.  ( {
a ,  b } 
\  { b } ) n  e.  ( G NeighbVtx  b ) ) )
67 difprsn1 4099 . . . . . . . . . . . . 13  |-  ( a  =/=  b  ->  ( { a ,  b }  \  { a } )  =  {
b } )
6867raleqdv 2979 . . . . . . . . . . . 12  |-  ( a  =/=  b  ->  ( A. n  e.  ( { a ,  b }  \  { a } ) n  e.  ( G NeighbVtx  a )  <->  A. n  e.  { b } n  e.  ( G NeighbVtx  a ) ) )
69 eleq1 2537 . . . . . . . . . . . . 13  |-  ( n  =  b  ->  (
n  e.  ( G NeighbVtx  a )  <->  b  e.  ( G NeighbVtx  a ) ) )
7055, 69ralsn 4001 . . . . . . . . . . . 12  |-  ( A. n  e.  { b } n  e.  ( G NeighbVtx  a )  <->  b  e.  ( G NeighbVtx  a ) )
7168, 70syl6bb 269 . . . . . . . . . . 11  |-  ( a  =/=  b  ->  ( A. n  e.  ( { a ,  b }  \  { a } ) n  e.  ( G NeighbVtx  a )  <->  b  e.  ( G NeighbVtx  a ) ) )
72 difprsn2 4100 . . . . . . . . . . . . 13  |-  ( a  =/=  b  ->  ( { a ,  b }  \  { b } )  =  {
a } )
7372raleqdv 2979 . . . . . . . . . . . 12  |-  ( a  =/=  b  ->  ( A. n  e.  ( { a ,  b }  \  { b } ) n  e.  ( G NeighbVtx  b )  <->  A. n  e.  { a } n  e.  ( G NeighbVtx  b ) ) )
74 eleq1 2537 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  (
n  e.  ( G NeighbVtx  b )  <->  a  e.  ( G NeighbVtx  b ) ) )
7554, 74ralsn 4001 . . . . . . . . . . . 12  |-  ( A. n  e.  { a } n  e.  ( G NeighbVtx  b )  <->  a  e.  ( G NeighbVtx  b ) )
7673, 75syl6bb 269 . . . . . . . . . . 11  |-  ( a  =/=  b  ->  ( A. n  e.  ( { a ,  b }  \  { b } ) n  e.  ( G NeighbVtx  b )  <->  a  e.  ( G NeighbVtx  b ) ) )
7771, 76anbi12d 725 . . . . . . . . . 10  |-  ( a  =/=  b  ->  (
( A. n  e.  ( { a ,  b }  \  {
a } ) n  e.  ( G NeighbVtx  a )  /\  A. n  e.  ( { a ,  b }  \  {
b } ) n  e.  ( G NeighbVtx  b ) )  <->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b ) ) ) )
7866, 77syl5bb 265 . . . . . . . . 9  |-  ( a  =/=  b  ->  ( A. v  e.  { a ,  b } A. n  e.  ( {
a ,  b } 
\  { v } ) n  e.  ( G NeighbVtx  v )  <->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b ) ) ) )
7953, 78sylan9bbr 715 . . . . . . . 8  |-  ( ( a  =/=  b  /\  V  =  { a ,  b } )  ->  ( A. v  e.  V  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v )  <-> 
( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b )
) ) )
8049, 79bibi12d 328 . . . . . . 7  |-  ( ( a  =/=  b  /\  V  =  { a ,  b } )  ->  ( ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) )  <->  ( { a ,  b }  e.  E 
<->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b )
) ) ) )
8180adantl 473 . . . . . 6  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) )  <->  ( {
a ,  b }  e.  E  <->  ( b  e.  ( G NeighbVtx  a )  /\  a  e.  ( G NeighbVtx  b ) ) ) ) )
8247, 81mpbird 240 . . . . 5  |-  ( ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  /\  ( a  =/=  b  /\  V  =  { a ,  b } ) )  -> 
( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) ) )
8382ex 441 . . . 4  |-  ( ( G  e. UHGraph  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( (
a  =/=  b  /\  V  =  { a ,  b } )  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) ) ) )
8483rexlimdvva 2878 . . 3  |-  ( G  e. UHGraph  ->  ( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  V  =  { a ,  b } )  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) ) )
855, 84syl5bi 225 . 2  |-  ( G  e. UHGraph  ->  ( ( # `  V )  =  2  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) ) ) )
8685imp 436 1  |-  ( ( G  e. UHGraph  /\  ( # `
 V )  =  2 )  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   2c2 10681   #chash 12553  Vtxcvtx 39251   UHGraph cuhgr 39300  Edgcedga 39371   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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
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-reu 2763  df-rmo 2764  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-uhgr 39302  df-edga 39372  df-nbgr 39565
This theorem is referenced by:  uvtx2vtx1edgb  39636
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