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Theorem 1vgrex 28809
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 6930 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2847 1 (𝑁𝑉𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3470  cfv 6543  Vtxcvtx 28803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-dm 5683  df-iota 6495  df-fv 6551
This theorem is referenced by:  upgr1e  28920  uspgr1e  29051  nbgrval  29143  cplgr1vlem  29236  vtxdgval  29276  vtxdgelxnn0  29280  wlkson  29464  trlsonfval  29514  pthsonfval  29548  spthson  29549  2wlkd  29741  is0wlk  29921  0wlkon  29924  is0trl  29927  0trlon  29928  0pthon  29931  0clwlkv  29935  1wlkd  29945  3wlkd  29974  wlkl0  30171
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