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Theorem usgr0e 26706
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 6521 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 usgr0e.g . . 3 (𝜑𝐺𝑊)
4 eqid 2795 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2795 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5isusgr 26626 . . 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 258 1 (𝜑𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1522  wcel 2081  {crab 3109  cdif 3860  c0 4215  𝒫 cpw 4457  {csn 4476  dom cdm 5448  1-1wf1 6227  cfv 6230  2c2 11545  chash 13545  Vtxcvtx 26469  iEdgciedg 26470  USGraphcusgr 26622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fv 6238  df-usgr 26624
This theorem is referenced by:  usgr0vb  26707  uhgr0vusgr  26712  usgr0eop  26716  edg0usgr  26723  usgr1v  26726  griedg0ssusgr  26735  cusgr1v  26901  frgr0v  27737
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