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| Mirrors > Home > MPE Home > Th. List > pwid | Structured version Visualization version GIF version | ||
| Description: A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
| Ref | Expression |
|---|---|
| pwid.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| pwid | ⊢ 𝐴 ∈ 𝒫 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pwid.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | pwidg 4620 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: pwnex 7779 r1ordg 9818 rankr1id 9902 cfss 10305 0ram 17058 evl1fval1lem 22334 bastg 22973 fincmp 23401 restlly 23491 ptbasfi 23589 zfbas 23904 ustfilxp 24221 minveclem3b 25462 wilthlem3 27113 coinflipprob 34482 mapdunirnN 41652 pwtrrVD 44845 vsetrec 49222 |
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