MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgr0eop Structured version   Visualization version   GIF version

Theorem usgr0eop 29079
Description: The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
Assertion
Ref Expression
usgr0eop (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)

Proof of Theorem usgr0eop
StepHypRef Expression
1 opex 5470 . . 3 𝑉, ∅⟩ ∈ V
21a1i 11 . 2 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3 0ex 5311 . . 3 ∅ ∈ V
4 opiedgfv 28840 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
53, 4mpan2 689 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
62, 5usgr0e 29069 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326  cop 4638  cfv 6553  iEdgciedg 28830  USGraphcusgr 28982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fv 6561  df-2nd 8000  df-iedg 28832  df-usgr 28984
This theorem is referenced by:  rgrusgrprc  29423
  Copyright terms: Public domain W3C validator