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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 2821 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 26847 | . . 3 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | pweq 4555 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅) | |
5 | 4 | difeq1d 4098 | . . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅})) |
6 | pw0 4745 | . . . . . . . . 9 ⊢ 𝒫 ∅ = {∅} | |
7 | 6 | difeq1i 4095 | . . . . . . . 8 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
8 | difid 4330 | . . . . . . . 8 ⊢ ({∅} ∖ {∅}) = ∅ | |
9 | 7, 8 | eqtri 2844 | . . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
10 | 5, 9 | syl6eq 2872 | . . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
12 | 11 | feq3d 6501 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)) |
13 | f00 6561 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅)) | |
14 | 13 | simplbi 500 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅) |
15 | 12, 14 | syl6bi 255 | . . 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 485 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
18 | simpr 487 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
19 | 17, 18 | uhgr0e 26856 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph) |
20 | 19 | ex 415 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
21 | 20 | adantr 483 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
22 | 16, 21 | impbid 214 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4567 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 UHGraphcuhgr 26841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-uhgr 26843 |
This theorem is referenced by: usgr0vb 27019 uhgr0v0e 27020 0uhgrsubgr 27061 finsumvtxdg2size 27332 0uhgrrusgr 27360 frgr0v 28041 frgruhgr0v 28043 |
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