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Theorem pwunim 4014
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 4012 . . 3 ((AB BA) → 𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B))
2 pwunss 4011 . . . 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 2954 . 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 628   = wceq 1242  cun 2909  wss 2911  𝒫 cpw 3351
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by: (None)
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