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Mirrors > Home > MPE Home > Th. List > pwun | Structured version Visualization version GIF version |
Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.) |
Ref | Expression |
---|---|
pwun | ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwunss 5446 | . . 3 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | |
2 | 1 | biantru 530 | . 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))) |
3 | pwssun 5448 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) | |
4 | eqss 3979 | . 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))) | |
5 | 2, 3, 4 | 3bitr4i 304 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∪ cun 3931 ⊆ wss 3933 𝒫 cpw 4535 |
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-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 |
This theorem is referenced by: (None) |
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