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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-discrmoore | Structured version Visualization version GIF version |
Description: The powerclass 𝒫 𝐴 is a Moore collection if and only if 𝐴 is a set. It is then called the discrete Moore collection. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-discrmoore | ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unipw 5309 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | ineq1i 4099 | . . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥) |
3 | inex1g 5187 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V) | |
4 | inss1 4119 | . . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴) |
6 | 3, 5 | elpwd 4496 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
7 | 2, 6 | eqeltrid 2837 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
9 | 8 | bj-ismooredr 34901 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore) |
10 | pwexr 7506 | . 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V) | |
11 | 9, 10 | impbii 212 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2114 Vcvv 3398 ∩ cin 3842 ⊆ wss 3843 𝒫 cpw 4488 ∪ cuni 4796 ∩ cint 4836 Moorecmoore 34895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-pw 4490 df-sn 4517 df-pr 4519 df-uni 4797 df-int 4837 df-bj-moore 34896 |
This theorem is referenced by: (None) |
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