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Theorem pwunss 4268
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 3306 . . 3 ((𝑥𝐴𝑥𝐵) → 𝑥 ⊆ (𝐴𝐵))
2 elun 3268 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 vex 2733 . . . . . 6 𝑥 ∈ V
43elpw 3572 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3572 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
64, 5orbi12i 759 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
72, 6bitri 183 . . 3 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
83elpw 3572 . . 3 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
91, 7, 83imtr4i 200 . 2 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴𝐵))
109ssriv 3151 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 703  wcel 2141  cun 3119  wss 3121  𝒫 cpw 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568
This theorem is referenced by:  pwundifss  4270  pwunim  4271
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