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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 5241 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
3 | 2 | bicomi 226 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 𝒫 cpw 4532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-pw 4534 |
This theorem is referenced by: (None) |
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