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Mirrors > Home > MPE Home > Th. List > elpwb | Structured version Visualization version GIF version |
Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
elpwb | ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3498 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V) | |
2 | elpwg 4607 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | biadanii 822 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-ss 3979 df-pw 4606 |
This theorem is referenced by: elpwpw 5106 elpwpwel 7785 onsupcl2 43213 onsupuni2 43218 onsupintrab2 43220 onuniintrab2 43223 |
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