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Theorem usgr0e 40462
 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 6082 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
3 usgr0e.g . . 3 (𝜑𝐺𝑊)
4 eqid 2610 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5isusgr 40383 . . 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 246 1 (𝜑𝐺 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-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-usgr 40381 This theorem is referenced by:  usgr0vb  40463  uhgr0vusgr  40468  usgr0eop  40472  edg0usgr  40479  usgr1v  40482  griedg0ssusgr  40489  cusgr1v  40653  frgr0v  41433
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