| Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
| 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 5334 | . 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 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |