ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwssunim Structured version   GIF version

Theorem pwssunim 4012
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 3110 . . . . 5 (BA ↔ (AB) = A)
2 pweq 3354 . . . . . 6 ((AB) = A → 𝒫 (AB) = 𝒫 A)
3 eqimss 2991 . . . . . 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 3107 . . . . 5 (AB ↔ (AB) = B)
7 pweq 3354 . . . . . 6 ((AB) = B → 𝒫 (AB) = 𝒫 B)
8 eqimss 2991 . . . . . 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 675 . . 3 ((BA AB) → (𝒫 (AB) ⊆ 𝒫 A 𝒫 (AB) ⊆ 𝒫 B))
1211orcoms 648 . 2 ((AB BA) → (𝒫 (AB) ⊆ 𝒫 A 𝒫 (AB) ⊆ 𝒫 B))
13 ssun 3116 . 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 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:  pwunim  4014
  Copyright terms: Public domain W3C validator