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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 6819 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
3 | usgr0e.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | eqid 2733 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2733 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | isusgr 28146 | . . 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 205 = wceq 1542 ∈ wcel 2107 {crab 3406 ∖ cdif 3908 ∅c0 4283 𝒫 cpw 4561 {csn 4587 dom cdm 5634 –1-1→wf1 6494 ‘cfv 6497 2c2 12213 ♯chash 14236 Vtxcvtx 27989 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 |
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-clab 2711 df-cleq 2725 df-clel 2811 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-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-usgr 28144 |
This theorem is referenced by: usgr0vb 28227 uhgr0vusgr 28232 usgr0eop 28236 edg0usgr 28243 usgr1v 28246 griedg0ssusgr 28255 cusgr1v 28421 frgr0v 29248 |
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