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Theorem usgr0eop 27034
 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 5333 . . 3 𝑉, ∅⟩ ∈ V
21a1i 11 . 2 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3 0ex 5187 . . 3 ∅ ∈ V
4 opiedgfv 26798 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
53, 4mpan2 690 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
62, 5usgr0e 27024 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114  Vcvv 3469  ∅c0 4265  ⟨cop 4545  ‘cfv 6334  iEdgciedg 26788  USGraphcusgr 26940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fv 6342  df-2nd 7676  df-iedg 26790  df-usgr 26942 This theorem is referenced by:  rgrusgrprc  27377
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