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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 3512 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V) | |
2 | elpwg 4544 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | biadanii 820 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 𝒫 cpw 4538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 |
This theorem is referenced by: elpwpw 5016 elpwpwel 7483 |
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