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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 5461 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | ineq1i 4224 | . . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥) |
3 | inex1g 5325 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V) | |
4 | inss1 4245 | . . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴) |
6 | 3, 5 | elpwd 4611 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
7 | 2, 6 | eqeltrid 2843 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
9 | 8 | bj-ismooredr 37092 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore) |
10 | pwexr 7784 | . 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V) | |
11 | 9, 10 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 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 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-int 4952 df-bj-moore 37087 |
This theorem is referenced by: (None) |
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