Theorem 1vgrex 29613

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

Step	Hyp	Ref	Expression
1		elfvex 6946	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2		1vgrex.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eleq2s 2882	1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ‘cfv 6561  Vtxcvtx 29607

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-dm 5666  df-iota 6515  df-fv 6569

This theorem is used by:  upgr1e  29724  uspgr1e  29858  nbgrval  29950  cplgr1vlem  30043  vtxdgval  30082  vtxdgelxnn0  30086  wlkson  30268  trlsonfval  30321  pthsonfval  30359  spthson  30360  2wlkd  30558  is0wlk  30741  0wlkon  30744  is0trl  30747  0trlon  30748  0pthon  30751  0clwlkv  30755  1wlkd  30765  3wlkd  30804  wlkl0  31001  clnbgrval  49135  isgrtri  49256