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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 37088 | . 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
2 | 0elpw 5362 | . . 3 ⊢ ∅ ∈ 𝒫 𝐴 | |
3 | rint0 4993 | . . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
4 | 3 | eleq1d 2824 | . . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) |
5 | 4 | rspcv 3618 | . . 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 217 | 1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 ∩ cint 4951 Moorecmoore 37086 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-uni 4913 df-int 4952 df-bj-moore 37087 |
This theorem is referenced by: bj-0nmoore 37095 |
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