Theorem usgr0e 29293

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 6806	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		usgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	isusgr 29210	. . 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 257	1 ⊢ (𝜑 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {crab 3390   ∖ cdif 3887  ∅c0 4274  𝒫 cpw 4542  {csn 4568  dom cdm 5622  –1-1→wf1 6487  ‘cfv 6490  2c2 12201  ♯chash 14254  Vtxcvtx 29053  iEdgciedg 29054  USGraphcusgr 29206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fv 6498  df-usgr 29208

This theorem is referenced by:  usgr0vb  29294  uhgr0vusgr  29299  usgr0eop  29303  edg0usgr  29310  usgr1v  29313  griedg0ssusgr  29322  cusgr1v  29488  frgr0v  30321