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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismoored | Structured version Visualization version GIF version |
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-ismoored.1 | ⊢ (𝜑 → 𝐴 ∈ Moore) |
bj-ismoored.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
bj-ismoored | ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteq 4878 | . . . 4 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵) | |
2 | 1 | ineq2d 4188 | . . 3 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵)) |
3 | 2 | eleq1d 2897 | . 2 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)) |
4 | bj-ismoored.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Moore) | |
5 | bj-ismoore 34396 | . . 3 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
6 | 4, 5 | sylib 220 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
7 | bj-ismoored.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
8 | 4, 7 | sselpwd 5229 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴) |
9 | 3, 6, 8 | rspcdva 3624 | 1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 ∩ cint 4875 Moorecmoore 34394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-uni 4838 df-int 4876 df-bj-moore 34395 |
This theorem is referenced by: bj-ismoored2 34399 |
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