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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 6882 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
| 3 | usgr0e.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 5 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 6 | 4, 5 | isusgr 29170 | . . 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 257 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 dom cdm 5685 –1-1→wf1 6558 ‘cfv 6561 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 USGraphcusgr 29166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 df-usgr 29168 |
| This theorem is referenced by: usgr0vb 29254 uhgr0vusgr 29259 usgr0eop 29263 edg0usgr 29270 usgr1v 29273 griedg0ssusgr 29282 cusgr1v 29448 frgr0v 30281 |
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