Theorem usgr0eop 29337

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.)

Assertion

Ref	Expression
usgr0eop	⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)

Proof of Theorem usgr0eop

Step	Hyp	Ref	Expression
1		opex 5421	. . 3 ⊢ ⟨𝑉, ∅⟩ ∈ V
2	1	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3		0ex 5256	. . 3 ⊢ ∅ ∈ V
4		opiedgfv 29098	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
5	3, 4	mpan2 692	. 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
6	2, 5	usgr0e 29327	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ⟨cop 4588  ‘cfv 6502  iEdgciedg 29088  USGraphcusgr 29240

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fv 6510  df-2nd 7946  df-iedg 29090  df-usgr 29242

This theorem is referenced by:  rgrusgrprc  29681