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Mirrors > Home > ILE Home > Th. List > elpw2 | GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.) |
Ref | Expression |
---|---|
elpw2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elpw2 | ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw2.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elpw2g 4142 | . 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 𝒫 cpw 3566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3568 |
This theorem is referenced by: elpwi2 4144 axpweq 4157 genpelxp 7473 ltexprlempr 7570 recexprlempr 7594 cauappcvgprlemcl 7615 cauappcvgprlemladd 7620 caucvgprlemcl 7638 caucvgprprlemcl 7666 uzf 9490 ixxf 9855 fzf 9969 cncfval 13353 reldvg 13442 dvfvalap 13444 |
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