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Mirrors > Home > MPE Home > Th. List > elpwpw | Structured version Visualization version GIF version |
Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
elpwpw | ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwb 4568 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵)) | |
2 | sspwuni 5060 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
3 | 2 | anbi2i 623 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) |
4 | 1, 3 | bitri 274 | 1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 𝒫 cpw 4560 ∪ cuni 4865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-v 3447 df-in 3917 df-ss 3927 df-pw 4562 df-uni 4866 |
This theorem is referenced by: pwpwab 5063 elpwpwel 7700 ismnushort 42562 |
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