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Theorem 1vgrex 26103
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 6362 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2868 1 (𝑁𝑉𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cfv 6031  Vtxcvtx 26095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-dm 5259  df-iota 5994  df-fv 6039
This theorem is referenced by:  upgr1e  26229  uspgr1e  26359  nbgrval  26452  cplgr1vlem  26560  vtxdgval  26599  vtxdgelxnn0  26603  wlkson  26787  trlsonfval  26837  pthsonfval  26871  spthson  26872  2wlkd  27083  is0wlk  27297  0wlkon  27300  is0trl  27303  0trlon  27304  0pthon  27307  0clwlkv  27311  1wlkd  27321  3wlkd  27350  wlkl0  27558
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