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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-discrmoore | Structured version Visualization version GIF version |
Description: The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-discrmoore | ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4850 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
2 | unipw 4918 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | ineq1i 3810 | . . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥) |
4 | inex1g 4801 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V) | |
5 | inss1 3833 | . . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴 | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴) |
7 | 4, 6 | elpwd 4167 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
8 | 3, 7 | syl5eqel 2705 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴) |
10 | 1, 9 | bj-ismooredr 33064 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore) |
11 | pwexr 6974 | . 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V) | |
12 | 10, 11 | impbii 199 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ∩ cint 4475 Moorecmoore 33057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-bj-moore 33058 |
This theorem is referenced by: (None) |
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