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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 4121 | . 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 1977 Vcvv 3173 𝒫 cpw 4108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 |
This theorem is referenced by: r1ordg 8524 rankr1id 8608 cfss 8970 0ram 15562 evl1fval1lem 19515 bastg 20581 fincmp 21006 restlly 21096 ptbasfi 21194 zfbas 21510 ustfilxp 21826 metustfbas 22172 minveclem3b 23007 wilthlem3 24596 coinflipprob 29868 bj-pwnex 32246 mapdunirnN 35957 pwtrrVD 38082 vsetrec 42245 |
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