Theorem usgr0e 29847

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	⊢ (𝜑 → 𝐺 ∈ 𝑊)
usgr0e.e	⊢ (𝜑 → (iEdg‘𝐺) = ∅)

Assertion

Ref	Expression
usgr0e	⊢ (𝜑 → 𝐺 ∈ USGraph)

Proof of Theorem usgr0e

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgr0e.e	. . 3 ⊢ (𝜑 → (iEdg‘𝐺) = ∅)
2	1	f10d 6882	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶1-1{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		usgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	isusgr 29764	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶1-1{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
7	3, 6	syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶1-1{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
8	2, 7	mpbird 260	1 ⊢ (𝜑 → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {crab 3416   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  {csn 4588  dom cdm 5655  ⟶1-1wf1 6555  ‘cfv 6558  2c2 12424  ♯chash 14501  Vtxcvtx 29604  iEdgciedg 29605  USGraphcusgr 29760

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fv 6566  df-usgr 29762

This theorem is used by:  usgr0vb  29848  uhgr0vusgr  29853  usgr0eop  29857  edg0usgr  29864  usgr1v  29867  griedg0ssusgr  29876  cusgr1v  30042  frgr0v  30893