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Mirrors > Home > MPE Home > Th. List > usgr0eop | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
usgr0eop | ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5422 | . . 3 ⊢ ⟨𝑉, ∅⟩ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ V) |
3 | 0ex 5265 | . . 3 ⊢ ∅ ∈ V | |
4 | opiedgfv 28000 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘⟨𝑉, ∅⟩) = ∅) | |
5 | 3, 4 | mpan2 690 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ∅⟩) = ∅) |
6 | 2, 5 | usgr0e 28226 | 1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 ⟨cop 4593 ‘cfv 6497 iEdgciedg 27990 USGraphcusgr 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fv 6505 df-2nd 7923 df-iedg 27992 df-usgr 28144 |
This theorem is referenced by: rgrusgrprc 28579 |
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