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Theorem usgr0eop 40472
 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 4859 . . 3 𝑉, ∅⟩ ∈ V
21a1i 11 . 2 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3 0ex 4718 . . 3 ∅ ∈ V
4 opiedgfv 25684 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
53, 4mpan2 703 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
62, 5usgr0e 40462 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  iEdgciedg 25674   USGraph cusgr 40379 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-2nd 7060  df-iedg 25676  df-usgr 40381 This theorem is referenced by:  rgrusgrprc  40789
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