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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 4561 | . 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 2114 Vcvv 3494 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-pw 4541 |
This theorem is referenced by: pwnex 7481 r1ordg 9207 rankr1id 9291 cfss 9687 0ram 16356 evl1fval1lem 20493 bastg 21574 fincmp 22001 restlly 22091 ptbasfi 22189 zfbas 22504 ustfilxp 22821 minveclem3b 24031 wilthlem3 25647 coinflipprob 31737 mapdunirnN 38801 pwtrrVD 41179 vsetrec 44825 |
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