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Theorem uvtxa0 39612
Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Hypothesis
Ref Expression
uvtxael.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
uvtxa0  |-  ( V  =  (/)  ->  (UnivVtx `  G
)  =  (/) )

Proof of Theorem uvtxa0
Dummy variables  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uvtxael.v . . . . . 6  |-  V  =  (Vtx `  G )
21uvtxaval 39605 . . . . 5  |-  ( G  e.  _V  ->  (UnivVtx `  G )  =  {
v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
32adantr 471 . . . 4  |-  ( ( G  e.  _V  /\  V  =  (/) )  -> 
(UnivVtx `  G )  =  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
4 rab0 3765 . . . . 5  |-  { v  e.  (/)  |  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) }  =  (/)
5 rabeq 3050 . . . . . . 7  |-  ( V  =  (/)  ->  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  =  { v  e.  (/)  | 
A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
65eqeq1d 2464 . . . . . 6  |-  ( V  =  (/)  ->  ( { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  =  (/)  <->  { v  e.  (/)  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  =  (/) ) )
76adantl 472 . . . . 5  |-  ( ( G  e.  _V  /\  V  =  (/) )  -> 
( { v  e.  V  |  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) }  =  (/)  <->  { v  e.  (/)  |  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) }  =  (/) ) )
84, 7mpbiri 241 . . . 4  |-  ( ( G  e.  _V  /\  V  =  (/) )  ->  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  =  (/) )
93, 8eqtrd 2496 . . 3  |-  ( ( G  e.  _V  /\  V  =  (/) )  -> 
(UnivVtx `  G )  =  (/) )
109ex 440 . 2  |-  ( G  e.  _V  ->  ( V  =  (/)  ->  (UnivVtx `  G )  =  (/) ) )
11 fvprc 5886 . . 3  |-  ( -.  G  e.  _V  ->  (UnivVtx `  G )  =  (/) )
1211a1d 26 . 2  |-  ( -.  G  e.  _V  ->  ( V  =  (/)  ->  (UnivVtx `  G )  =  (/) ) )
1310, 12pm2.61i 169 1  |-  ( V  =  (/)  ->  (UnivVtx `  G
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   {crab 2753   _Vcvv 3057    \ cdif 3413   (/)c0 3743   {csn 3980   ` cfv 5605  (class class class)co 6320  Vtxcvtx 39243   NeighbVtx cnbgr 39543  UnivVtxcuvtxa 39544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-uvtxa 39549
This theorem is referenced by: (None)
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