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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 33059 | . . 3 ⊢ (𝐴 ∈ Moore → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | |
2 | 0elpw 4834 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
3 | rint0 4517 | . . . . . 6 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
4 | 3 | eleq1d 2686 | . . . . 5 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) |
5 | 4 | rspcv 3305 | . . . 4 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴)) |
6 | 2, 5 | ax-mp 5 | . . 3 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴) |
7 | 1, 6 | syl6bi 243 | . 2 ⊢ (𝐴 ∈ Moore → (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)) |
8 | 7 | pm2.43i 52 | 1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 ∩ cint 4475 Moorecmoore 33057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-uni 4437 df-int 4476 df-bj-moore 33058 |
This theorem is referenced by: bj-0nmoore 33067 |
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