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Theorem usgra0 25899
 Description: The empty graph, with vertices but no edges, is a graph, analogous to umgra0 25854. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by AV, 25-Nov-2020.)
Assertion
Ref Expression
usgra0 (𝑉𝑊𝑉 USGrph ∅)

Proof of Theorem usgra0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2611 . . 3 (𝑉𝑊 → ∅ = ∅)
21f10d 6082 . 2 (𝑉𝑊 → ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
3 0ex 4718 . . 3 ∅ ∈ V
4 isusgra 25873 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (𝑉 USGrph ∅ ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
53, 4mpan2 703 . 2 (𝑉𝑊 → (𝑉 USGrph ∅ ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
62, 5mpbird 246 1 (𝑉𝑊𝑉 USGrph ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859 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-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-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-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-usgra 25862 This theorem is referenced by:  usgra0v  25900  usgra1v  25919  cusgra0v  25989  cusgra1v  25990
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