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Theorem 1vgrex 28262
Description: A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
Hypothesis
Ref Expression
1vgrex.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
1vgrex (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

Proof of Theorem 1vgrex
StepHypRef Expression
1 elfvex 6930 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2852 1 (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  â€˜cfv 6544  Vtxcvtx 28256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552
This theorem is referenced by:  upgr1e  28373  uspgr1e  28501  nbgrval  28593  cplgr1vlem  28686  vtxdgval  28725  vtxdgelxnn0  28729  wlkson  28913  trlsonfval  28963  pthsonfval  28997  spthson  28998  2wlkd  29190  is0wlk  29370  0wlkon  29373  is0trl  29376  0trlon  29377  0pthon  29380  0clwlkv  29384  1wlkd  29394  3wlkd  29423  wlkl0  29620
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