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Mirrors > Home > ILE Home > Th. List > elpwb | 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 2692 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V) | |
2 | elpwg 3513 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | biadan2 451 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 𝒫 cpw 3505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 |
This theorem is referenced by: elpwpw 3894 elpwpwel 4391 |
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