MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1vgrex Structured version   Visualization version   GIF version

Theorem 1vgrex 25782
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 (𝑁𝑉𝐺 ∈ V)

Proof of Theorem 1vgrex
StepHypRef Expression
1 elfvex 6178 . 2 (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
2 1vgrex.v . 2 𝑉 = (Vtx‘𝐺)
31, 2eleq2s 2716 1 (𝑁𝑉𝐺 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cfv 5847  Vtxcvtx 25774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855
This theorem is referenced by:  upgr1e  25903  uspgr1e  26029  usgr1e  26030  nbgrval  26121  dfnbgr3  26123  nbgr2vtx1edg  26133  uvtx2vtx1edg  26186  uvtxnbgrb  26189  cplgr1vlem  26212  vtxdgval  26251  vtxdgelxnn0  26254  wlkson  26421  trlsonfval  26471  pthsonfval  26505  spthson  26506  2wlkd  26701  is0wlk  26844  0wlkon  26847  is0trl  26850  0trlon  26851  0pthon  26854  1wlkd  26867  3wlkd  26896
  Copyright terms: Public domain W3C validator