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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismooredr | Structured version Visualization version GIF version |
Description: Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-ismooredr.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
bj-ismooredr.2 | ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
Ref | Expression |
---|---|
bj-ismooredr | ⊢ (𝜑 → 𝐴 ∈ Moore) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4426 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | bj-ismooredr.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
3 | 2 | ex 405 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
4 | 1, 3 | syl5 34 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
5 | 4 | ralrimiv 3124 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
6 | bj-ismooredr.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | bj-ismoore 33944 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
9 | 5, 8 | mpbird 249 | 1 ⊢ (𝜑 → 𝐴 ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2051 ∀wral 3081 ∩ cin 3821 ⊆ wss 3822 𝒫 cpw 4416 ∪ cuni 4708 ∩ cint 4745 Moorecmoore 33942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-in 3829 df-ss 3836 df-pw 4418 df-uni 4709 df-bj-moore 33943 |
This theorem is referenced by: bj-discrmoore 33951 |
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