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Theorem usgr0eop 26955
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 5347 . . 3 𝑉, ∅⟩ ∈ V
21a1i 11 . 2 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3 0ex 5202 . . 3 ∅ ∈ V
4 opiedgfv 26719 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
53, 4mpan2 687 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
62, 5usgr0e 26945 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288  cop 4563  cfv 6348  iEdgciedg 26709  USGraphcusgr 26861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356  df-2nd 7679  df-iedg 26711  df-usgr 26863
This theorem is referenced by:  rgrusgrprc  27298
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