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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 5129 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V) | |
2 | elssuni 4738 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
3 | sspwb 5195 | . . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) | |
4 | 2, 3 | sylib 210 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) |
5 | 1, 4 | elpwd 4426 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 Vcvv 3410 ⊆ wss 3824 𝒫 cpw 4417 ∪ cuni 4709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-pw 4419 df-sn 4437 df-pr 4439 df-uni 4710 |
This theorem is referenced by: (None) |
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