Theorem 1vgrex 28991

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 6866	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2		1vgrex.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eleq2s 2851	1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ‘cfv 6489  Vtxcvtx 28985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-dm 5631  df-iota 6445  df-fv 6497

This theorem is referenced by:  upgr1e  29102  uspgr1e  29233  nbgrval  29325  cplgr1vlem  29418  vtxdgval  29458  vtxdgelxnn0  29462  wlkson  29644  trlsonfval  29693  pthsonfval  29729  spthson  29730  2wlkd  29925  is0wlk  30108  0wlkon  30111  is0trl  30114  0trlon  30115  0pthon  30118  0clwlkv  30122  1wlkd  30132  3wlkd  30161  wlkl0  30358  clnbgrval  47936  isgrtri  48057