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Theorem usgr0e 28226
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 6819 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 usgr0e.g . . 3 (𝜑𝐺𝑊)
4 eqid 2733 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2733 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5isusgr 28146 . . 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 205   = wceq 1542  wcel 2107  {crab 3406  cdif 3908  c0 4283  𝒫 cpw 4561  {csn 4587  dom cdm 5634  1-1wf1 6494  cfv 6497  2c2 12213  chash 14236  Vtxcvtx 27989  iEdgciedg 27990  USGraphcusgr 28142
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505  df-usgr 28144
This theorem is referenced by:  usgr0vb  28227  uhgr0vusgr  28232  usgr0eop  28236  edg0usgr  28243  usgr1v  28246  griedg0ssusgr  28255  cusgr1v  28421  frgr0v  29248
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