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Theorem pwunim 3997
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 ((AB BA) → 𝒫 (AB) = (𝒫 A ∪ 𝒫 B))

Proof of Theorem pwunim
StepHypRef Expression
1 pwssunim 3995 . . 3 ((AB BA) → 𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B))
2 pwunss 3994 . . . 4 (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)
32biantru 286 . . 3 (𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B) ↔ (𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B) (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)))
41, 3sylib 127 . 2 ((AB BA) → (𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B) (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)))
5 eqss 2937 . 2 (𝒫 (AB) = (𝒫 A ∪ 𝒫 B) ↔ (𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B) (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)))
64, 5sylibr 137 1 ((AB BA) → 𝒫 (AB) = (𝒫 A ∪ 𝒫 B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   = wceq 1228  cun 2892  wss 2894  𝒫 cpw 3334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336
This theorem is referenced by: (None)
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