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Theorem pwun 5530
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.)
Assertion
Ref Expression
pwun ((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))

Proof of Theorem pwun
StepHypRef Expression
1 pwunss 4579 . . 3 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
21biantru 531 . 2 (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
3 pwssun 5529 . 2 ((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
4 eqss 3960 . 2 (𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
52, 3, 43bitr4i 303 1 ((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  cun 3909  wss 3911  𝒫 cpw 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-un 3916  df-in 3918  df-ss 3928  df-pw 4563  df-sn 4588  df-pr 4590
This theorem is referenced by: (None)
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