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Theorem 1vgrex 26798
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 6694 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2934 1 (𝑁𝑉𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cfv 6343  Vtxcvtx 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196  ax-pow 5253
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-dm 5552  df-iota 6302  df-fv 6351
This theorem is referenced by:  upgr1e  26909  uspgr1e  27037  nbgrval  27129  cplgr1vlem  27222  vtxdgval  27261  vtxdgelxnn0  27265  wlkson  27449  trlsonfval  27498  pthsonfval  27532  spthson  27533  2wlkd  27725  is0wlk  27905  0wlkon  27908  is0trl  27911  0trlon  27912  0pthon  27915  0clwlkv  27919  1wlkd  27929  3wlkd  27958  wlkl0  28155
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