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Theorem pwssunim 3995
Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwssunim ((AB BA) → 𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B))

Proof of Theorem pwssunim
StepHypRef Expression
1 ssequn2 3093 . . . . 5 (BA ↔ (AB) = A)
2 pweq 3337 . . . . . 6 ((AB) = A → 𝒫 (AB) = 𝒫 A)
3 eqimss 2974 . . . . . 6 (𝒫 (AB) = 𝒫 A → 𝒫 (AB) ⊆ 𝒫 A)
42, 3syl 14 . . . . 5 ((AB) = A → 𝒫 (AB) ⊆ 𝒫 A)
51, 4sylbi 114 . . . 4 (BA → 𝒫 (AB) ⊆ 𝒫 A)
6 ssequn1 3090 . . . . 5 (AB ↔ (AB) = B)
7 pweq 3337 . . . . . 6 ((AB) = B → 𝒫 (AB) = 𝒫 B)
8 eqimss 2974 . . . . . 6 (𝒫 (AB) = 𝒫 B → 𝒫 (AB) ⊆ 𝒫 B)
97, 8syl 14 . . . . 5 ((AB) = B → 𝒫 (AB) ⊆ 𝒫 B)
106, 9sylbi 114 . . . 4 (AB → 𝒫 (AB) ⊆ 𝒫 B)
115, 10orim12i 663 . . 3 ((BA AB) → (𝒫 (AB) ⊆ 𝒫 A 𝒫 (AB) ⊆ 𝒫 B))
1211orcoms 636 . 2 ((AB BA) → (𝒫 (AB) ⊆ 𝒫 A 𝒫 (AB) ⊆ 𝒫 B))
13 ssun 3099 . 2 ((𝒫 (AB) ⊆ 𝒫 A 𝒫 (AB) ⊆ 𝒫 B) → 𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B))
1412, 13syl 14 1 ((AB BA) → 𝒫 (AB) ⊆ (𝒫 A ∪ 𝒫 B))
Colors of variables: wff set class
Syntax hints:  wi 4   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:  pwunim  3997
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