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Theorem usgr0e 29253
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.) (Proof shortened by AV, 25-Nov-2020.)
Hypotheses
Ref Expression
usgr0e.g (𝜑𝐺𝑊)
usgr0e.e (𝜑 → (iEdg‘𝐺) = ∅)
Assertion
Ref Expression
usgr0e (𝜑𝐺 ∈ USGraph)

Proof of Theorem usgr0e
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 usgr0e.e . . 3 (𝜑 → (iEdg‘𝐺) = ∅)
21f10d 6882 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 usgr0e.g . . 3 (𝜑𝐺𝑊)
4 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5isusgr 29170 . . 3 (𝐺𝑊 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
73, 6syl 17 . 2 (𝜑 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
82, 7mpbird 257 1 (𝜑𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {crab 3436  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  1-1wf1 6558  cfv 6561  2c2 12321  chash 14369  Vtxcvtx 29013  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
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-clab 2715  df-cleq 2729  df-clel 2816  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-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-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-usgr 29168
This theorem is referenced by:  usgr0vb  29254  uhgr0vusgr  29259  usgr0eop  29263  edg0usgr  29270  usgr1v  29273  griedg0ssusgr  29282  cusgr1v  29448  frgr0v  30281
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