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Theorem pwunss 5304
 Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwunss (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)

Proof of Theorem pwunss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssun 4052 . . 3 ((𝑥𝐴𝑥𝐵) → 𝑥 ⊆ (𝐴𝐵))
2 elun 4013 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 selpw 4427 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
4 selpw 4427 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
53, 4orbi12i 898 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
62, 5bitri 267 . . 3 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
7 selpw 4427 . . 3 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
81, 6, 73imtr4i 284 . 2 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴𝐵))
98ssriv 3861 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 833   ∈ wcel 2048   ∪ cun 3826   ⊆ wss 3828  𝒫 cpw 4420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2747 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-v 3414  df-un 3833  df-in 3835  df-ss 3842  df-pw 4422 This theorem is referenced by:  pwundif  5306  pwun  5307
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