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Mirrors > Home > MPE Home > Th. List > usgr0e | 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.) (Proof shortened by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
usgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
usgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) |
Ref | Expression |
---|---|
usgr0e | ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0e.e | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
2 | 1 | f10d 6332 | . 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 26268 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
8 | 2, 7 | mpbird 247 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1632 ∈ wcel 2139 {crab 3054 ∖ cdif 3712 ∅c0 4058 𝒫 cpw 4302 {csn 4321 dom cdm 5266 –1-1→wf1 6046 ‘cfv 6049 2c2 11282 ♯chash 13331 Vtxcvtx 26094 iEdgciedg 26095 USGraphcusgr 26264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fv 6057 df-usgr 26266 |
This theorem is referenced by: usgr0vb 26349 uhgr0vusgr 26354 usgr0eop 26358 edg0usgr 26365 usgr1v 26368 griedg0ssusgr 26377 cusgr1v 26558 frgr0v 27436 |
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