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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 35986 | . 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
2 | 0elpw 5355 | . . 3 ⊢ ∅ ∈ 𝒫 𝐴 | |
3 | rint0 4995 | . . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
4 | 3 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) |
5 | 4 | rspcv 3609 | . . 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 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∩ cint 4951 Moorecmoore 35984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-uni 4910 df-int 4952 df-bj-moore 35985 |
This theorem is referenced by: bj-0nmoore 35993 |
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