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Theorem uvtxisvtx 29424
Description: A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.)
Hypothesis
Ref Expression
uvtxel.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uvtxisvtx (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁𝑉)

Proof of Theorem uvtxisvtx
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 uvtxel.v . . 3 𝑉 = (Vtx‘𝐺)
21uvtxel 29423 . 2 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
32simplbi 497 1 (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  wral 3067  cdif 3973  {csn 4648  cfv 6573  (class class class)co 7448  Vtxcvtx 29031   NeighbVtx cnbgr 29367  UnivVtxcuvtx 29420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-uvtx 29421
This theorem is referenced by:  uvtxssvtx  29425  uvtxnm1nbgr  29439  vdiscusgr  29567
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