Theorem usgr0e 29751

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 6847	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		usgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		eqid 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	isusgr 29668	. . 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 2145  {crab 3412   ∖ cdif 3895  ∅c0 4278  𝒫 cpw 4556  {csn 4583  dom cdm 5647  –1-1→wf1 6524  ‘cfv 6527  2c2 12367  ♯chash 14442  Vtxcvtx 29508  iEdgciedg 29509  USGraphcusgr 29664

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fv 6535  df-usgr 29666

This theorem is used by:  usgr0vb  29752  uhgr0vusgr  29757  usgr0eop  29761  edg0usgr  29768  usgr1v  29771  griedg0ssusgr  29780  cusgr1v  29946  frgr0v  30797