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Theorem pwunss 5169
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 3935 . . 3 ((𝑥𝐴𝑥𝐵) → 𝑥 ⊆ (𝐴𝐵))
2 elun 3896 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 selpw 4309 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
4 selpw 4309 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
53, 4orbi12i 544 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
62, 5bitri 264 . . 3 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
7 selpw 4309 . . 3 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
81, 6, 73imtr4i 281 . 2 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴𝐵))
98ssriv 3748 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wo 382  wcel 2139  cun 3713  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-pw 4304
This theorem is referenced by:  pwundif  5171  pwun  5172
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