Theorem 1vgrex 25679

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 6131	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2		1vgrex.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	eleq2s 2706	1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  Vtxcvtx 25673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812

This theorem is referenced by:  upgr1e  25779  uspgr1e  40470  usgr1e  40471  nbgrval  40560  dfnbgr3  40562  nbgr2vtx1edg  40572  uvtx2vtx1edg  40625  uvtxnbgrb  40628  cplgr1vlem  40651  vtxdgval  40684  vtxdgelxnn0  40687  wlkson  40864  trlsonfval  40913  pthsonfval  40946  spthson  40947  uhgr1wlkspth  40961  21wlkd  41143  is01wlk  41285  0wlkOn  41288  is0Trl  41291  0TrlOn  41292  0pthon-av  41295  11wlkd  41308  31wlkd  41337  eupth0  41382