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Mirrors > Home > MPE Home > Th. List > uhgr0vb | Structured version Visualization version GIF version |
Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgr0vb | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2818 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 26774 | . . 3 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | pweq 4538 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅) | |
5 | 4 | difeq1d 4095 | . . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅})) |
6 | pw0 4737 | . . . . . . . . 9 ⊢ 𝒫 ∅ = {∅} | |
7 | 6 | difeq1i 4092 | . . . . . . . 8 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
8 | difid 4327 | . . . . . . . 8 ⊢ ({∅} ∖ {∅}) = ∅ | |
9 | 7, 8 | eqtri 2841 | . . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
10 | 5, 9 | syl6eq 2869 | . . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
11 | 10 | adantl 482 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
12 | 11 | feq3d 6494 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)) |
13 | f00 6554 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅)) | |
14 | 13 | simplbi 498 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅) |
15 | 12, 14 | syl6bi 254 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺) = ∅)) |
16 | 3, 15 | syl5 34 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)) |
17 | simpl 483 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
18 | simpr 485 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
19 | 17, 18 | uhgr0e 26783 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph) |
20 | 19 | ex 413 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
21 | 20 | adantr 481 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
22 | 16, 21 | impbid 213 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∅c0 4288 𝒫 cpw 4535 {csn 4557 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 Vtxcvtx 26708 iEdgciedg 26709 UHGraphcuhgr 26768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-uhgr 26770 |
This theorem is referenced by: usgr0vb 26946 uhgr0v0e 26947 0uhgrsubgr 26988 finsumvtxdg2size 27259 0uhgrrusgr 27287 frgr0v 27968 frgruhgr0v 27970 |
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