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Theorem pwunim 4269
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 Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwunim ((𝐴𝐵𝐵𝐴) → 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))

Proof of Theorem pwunim
StepHypRef Expression
1 pwssunim 4267 . . 3 ((𝐴𝐵𝐵𝐴) → 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
2 pwunss 4266 . . . 4 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
32biantru 300 . . 3 (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
41, 3sylib 121 . 2 ((𝐴𝐵𝐵𝐴) → (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
5 eqss 3162 . 2 (𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)))
64, 5sylibr 133 1 ((𝐴𝐵𝐵𝐴) → 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 703   = wceq 1348  cun 3119  wss 3121  𝒫 cpw 3564
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 3566
This theorem is referenced by: (None)
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