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Mirrors > Home > MPE Home > Th. List > pwel | Structured version Visualization version GIF version |
Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
Ref | Expression |
---|---|
pwel | ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 5270 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V) | |
2 | elssuni 4859 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
3 | sspwb 5332 | . . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) | |
4 | 2, 3 | sylib 219 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) |
5 | 1, 4 | elpwd 4546 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-sn 4558 df-pr 4560 df-uni 4831 |
This theorem is referenced by: (None) |
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