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Theorem 1vgrex 29033
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 (𝑁𝑉𝐺 ∈ V)

Proof of Theorem 1vgrex
StepHypRef Expression
1 elfvex 6944 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2856 1 (𝑁𝑉𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cfv 6562  Vtxcvtx 29027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-dm 5698  df-iota 6515  df-fv 6570
This theorem is referenced by:  upgr1e  29144  uspgr1e  29275  nbgrval  29367  cplgr1vlem  29460  vtxdgval  29500  vtxdgelxnn0  29504  wlkson  29688  trlsonfval  29738  pthsonfval  29772  spthson  29773  2wlkd  29965  is0wlk  30145  0wlkon  30148  is0trl  30151  0trlon  30152  0pthon  30155  0clwlkv  30159  1wlkd  30169  3wlkd  30198  wlkl0  30395  clnbgrval  47746  isgrtri  47847
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