Theorem 1vgrex 29071

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 6875	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2		1vgrex.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eleq2s 2854	1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ‘cfv 6498  Vtxcvtx 29065

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-dm 5641  df-iota 6454  df-fv 6506

This theorem is referenced by:  upgr1e  29182  uspgr1e  29313  nbgrval  29405  cplgr1vlem  29498  vtxdgval  29537  vtxdgelxnn0  29541  wlkson  29723  trlsonfval  29772  pthsonfval  29808  spthson  29809  2wlkd  30004  is0wlk  30187  0wlkon  30190  is0trl  30193  0trlon  30194  0pthon  30197  0clwlkv  30201  1wlkd  30211  3wlkd  30240  wlkl0  30437  clnbgrval  48298  isgrtri  48419