Theorem usgr0e 28760

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 6866	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		usgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		eqid 2730	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2730	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	isusgr 28680	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
7	3, 6	syl 17	. 2 ⊢ (𝜑 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
8	2, 7	mpbird 256	1 ⊢ (𝜑 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  {crab 3430   ∖ cdif 3944  ∅c0 4321  𝒫 cpw 4601  {csn 4627  dom cdm 5675  –1-1→wf1 6539  ‘cfv 6542  2c2 12271  ♯chash 14294  Vtxcvtx 28523  iEdgciedg 28524  USGraphcusgr 28676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fv 6550  df-usgr 28678

This theorem is referenced by:  usgr0vb  28761  uhgr0vusgr  28766  usgr0eop  28770  edg0usgr  28777  usgr1v  28780  griedg0ssusgr  28789  cusgr1v  28955  frgr0v  29782