Theorem 1vgrex 25882

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 e. V -> G e. _V )

Proof of Theorem 1vgrex

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( N e. (Vtx ` G ) -> G e. _V )
2		1vgrex.v	. 2 |- V = (Vtx ` G )
3	1, 2	eleq2s 2719	1 |- ( N e. V -> G e. _V )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  Vtxcvtx 25874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by:  upgr1e  26008  uspgr1e  26136  nbgrval  26234  nbgr2vtx1edg  26246  uvtx2vtx1edg  26299  uvtxnbgrb  26302  cplgr1vlem  26325  vtxdgval  26364  vtxdgelxnn0  26368  wlkson  26552  trlsonfval  26602  pthsonfval  26636  spthson  26637  2wlkd  26832  is0wlk  26978  0wlkon  26981  is0trl  26984  0trlon  26985  0pthon  26988  1wlkd  27001  3wlkd  27030