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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 5239 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
3 | 2 | bicomi 225 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 |
This theorem is referenced by: (None) |
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