Theorem usgr0e 29372

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 6826	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		usgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		eqid 2752	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2752	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	isusgr 29289	. . 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 259	1 ⊢ (𝜑 → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1550   ∈ wcel 2132  {crab 3404   ∖ cdif 3892  ∅c0 4276  𝒫 cpw 4545  {csn 4572  dom cdm 5636  –1-1→wf1 6503  ‘cfv 6506  2c2 12258  ♯chash 14329  Vtxcvtx 29132  iEdgciedg 29133  USGraphcusgr 29285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-mo 2556  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fv 6514  df-usgr 29287

This theorem is referenced by:  usgr0vb  29373  uhgr0vusgr  29378  usgr0eop  29382  edg0usgr  29389  usgr1v  29392  griedg0ssusgr  29401  cusgr1v  29567  frgr0v  30399