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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwbi | Structured version Visualization version GIF version |
Description: Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.) |
Ref | Expression |
---|---|
elpwbi.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elpwbi | ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwbi.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | elpw2 5339 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
3 | 2 | bicomi 224 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ 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 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-pw 4606 |
This theorem is referenced by: (None) |
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