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Theorem usgr0eop 29263
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 5469 . . 3 𝑉, ∅⟩ ∈ V
21a1i 11 . 2 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3 0ex 5307 . . 3 ∅ ∈ V
4 opiedgfv 29024 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
53, 4mpan2 691 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
62, 5usgr0e 29253 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cfv 6561  iEdgciedg 29014  USGraphcusgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-2nd 8015  df-iedg 29016  df-usgr 29168
This theorem is referenced by:  rgrusgrprc  29607
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