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Theorem pwunss 4298
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 3329 . . 3 ((𝑥𝐴𝑥𝐵) → 𝑥 ⊆ (𝐴𝐵))
2 elun 3291 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 vex 2755 . . . . . 6 𝑥 ∈ V
43elpw 3596 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3596 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
64, 5orbi12i 765 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
72, 6bitri 184 . . 3 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
83elpw 3596 . . 3 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
91, 7, 83imtr4i 201 . 2 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴𝐵))
109ssriv 3174 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 709  wcel 2160  cun 3142  wss 3144  𝒫 cpw 3590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by:  pwundifss  4300  pwunim  4301
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