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Theorem pwun 5569
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 4617 . . 3 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
21biantru 529 . 2 (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
3 pwssun 5568 . 2 ((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
4 eqss 3994 . 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 395  wo 846   = wceq 1534  cun 3943  wss 3945  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-un 3950  df-in 3952  df-ss 3962  df-pw 4601  df-sn 4626  df-pr 4628
This theorem is referenced by: (None)
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