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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 5451 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
2 | 1 | ineq1i 4209 | . . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥) |
3 | inex1g 5320 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V) | |
4 | inss1 4229 | . . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴) |
6 | 3, 5 | elpwd 4609 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
7 | 2, 6 | eqeltrid 2836 | . . . 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 36294 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore) |
10 | pwexr 7755 | . 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V) | |
11 | 9, 10 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2105 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 ∩ cint 4951 Moorecmoore 36288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-int 4952 df-bj-moore 36289 |
This theorem is referenced by: (None) |
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