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Theorem usgr0e 27603
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 6750 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 usgr0e.g . . 3 (𝜑𝐺𝑊)
4 eqid 2738 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2738 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5isusgr 27523 . . 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 256 1 (𝜑𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2106  {crab 3068  cdif 3884  c0 4256  𝒫 cpw 4533  {csn 4561  dom cdm 5589  1-1wf1 6430  cfv 6433  2c2 12028  chash 14044  Vtxcvtx 27366  iEdgciedg 27367  USGraphcusgr 27519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-usgr 27521
This theorem is referenced by:  usgr0vb  27604  uhgr0vusgr  27609  usgr0eop  27613  edg0usgr  27620  usgr1v  27623  griedg0ssusgr  27632  cusgr1v  27798  frgr0v  28626
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