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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 5475 | . . 3 ⊢ 〈𝑉, ∅〉 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ V) |
3 | 0ex 5313 | . . 3 ⊢ ∅ ∈ V | |
4 | opiedgfv 29039 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘〈𝑉, ∅〉) = ∅) | |
5 | 3, 4 | mpan2 691 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ∅〉) = ∅) |
6 | 2, 5 | usgr0e 29268 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 ‘cfv 6563 iEdgciedg 29029 USGraphcusgr 29181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fv 6571 df-2nd 8014 df-iedg 29031 df-usgr 29183 |
This theorem is referenced by: rgrusgrprc 29622 |
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