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Mirrors > Home > MPE Home > Th. List > uhgra0v | 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.) |
Ref | Expression |
---|---|
uhgra0v | ⊢ (∅ UHGrph 𝐸 ↔ 𝐸 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrav 25825 | . . 3 ⊢ (∅ UHGrph 𝐸 → (∅ ∈ V ∧ 𝐸 ∈ V)) | |
2 | isuhgra 25827 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UHGrph 𝐸 ↔ 𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅}))) | |
3 | eqid 2610 | . . . . . 6 ⊢ dom 𝐸 = dom 𝐸 | |
4 | pw0 4283 | . . . . . . . 8 ⊢ 𝒫 ∅ = {∅} | |
5 | 4 | difeq1i 3686 | . . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
6 | difid 3902 | . . . . . . 7 ⊢ ({∅} ∖ {∅}) = ∅ | |
7 | 5, 6 | eqtri 2632 | . . . . . 6 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
8 | 3, 7 | feq23i 5952 | . . . . 5 ⊢ (𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅}) ↔ 𝐸:dom 𝐸⟶∅) |
9 | f00 6000 | . . . . . 6 ⊢ (𝐸:dom 𝐸⟶∅ ↔ (𝐸 = ∅ ∧ dom 𝐸 = ∅)) | |
10 | 9 | simplbi 475 | . . . . 5 ⊢ (𝐸:dom 𝐸⟶∅ → 𝐸 = ∅) |
11 | 8, 10 | sylbi 206 | . . . 4 ⊢ (𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅}) → 𝐸 = ∅) |
12 | 2, 11 | syl6bi 242 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UHGrph 𝐸 → 𝐸 = ∅)) |
13 | 1, 12 | mpcom 37 | . 2 ⊢ (∅ UHGrph 𝐸 → 𝐸 = ∅) |
14 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
15 | uhgra0 25838 | . . . 4 ⊢ (∅ ∈ V → ∅ UHGrph ∅) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ∅ UHGrph ∅ |
17 | breq2 4587 | . . 3 ⊢ (𝐸 = ∅ → (∅ UHGrph 𝐸 ↔ ∅ UHGrph ∅)) | |
18 | 16, 17 | mpbiri 247 | . 2 ⊢ (𝐸 = ∅ → ∅ UHGrph 𝐸) |
19 | 13, 18 | impbii 198 | 1 ⊢ (∅ UHGrph 𝐸 ↔ 𝐸 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 UHGrph cuhg 25819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-uhgra 25821 |
This theorem is referenced by: (None) |
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