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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismoored0 | Structured version Visualization version GIF version |
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-ismoored0 | ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ismoore 36289 | . 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
2 | 0elpw 5353 | . . 3 ⊢ ∅ ∈ 𝒫 𝐴 | |
3 | rint0 4993 | . . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
4 | 3 | eleq1d 2816 | . . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) |
5 | 4 | rspcv 3607 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∩ cin 3946 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ∩ cint 4949 Moorecmoore 36287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-in 3954 df-ss 3964 df-nul 4322 df-pw 4603 df-uni 4908 df-int 4950 df-bj-moore 36288 |
This theorem is referenced by: bj-0nmoore 36296 |
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