Theorem usgr0eop 29303

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 5405	. . 3 ⊢ ⟨𝑉, ∅⟩ ∈ V
2	1	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ V)
3		0ex 5231	. . 3 ⊢ ∅ ∈ V
4		opiedgfv 29064	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅)
5	3, 4	mpan2 692	. 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅)
6	2, 5	usgr0e 29293	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3427  ∅c0 4263  ⟨cop 4563  ‘cfv 6487  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-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fv 6495  df-2nd 7932  df-iedg 29056  df-usgr 29208

This theorem is referenced by:  rgrusgrprc  29646