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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 5455 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
| 2 | 1 | ineq1i 4216 | . . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥) |
| 3 | inex1g 5319 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V) | |
| 4 | inss1 4237 | . . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴 | |
| 5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴) |
| 6 | 3, 5 | elpwd 4606 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
| 7 | 2, 6 | eqeltrid 2845 | . . . 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 37110 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore) |
| 10 | pwexr 7785 | . 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 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 Moorecmoore 37104 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-bj-moore 37105 |
| This theorem is referenced by: (None) |
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