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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 | elssuni 4467 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
2 | sspwb 4917 | . . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) |
4 | pwexg 4850 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V) | |
5 | elpwg 4166 | . . 3 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)) |
7 | 3, 6 | mpbird 247 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 |
This theorem is referenced by: (None) |
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