Theorem pwssunim 4269

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	⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

Proof of Theorem pwssunim

Step	Hyp	Ref	Expression
1		ssequn2 3300	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
2		pweq 3569	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴)
3		eqimss 3201	. . . . . 6 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
4	2, 3	syl 14	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
5	1, 4	sylbi 120	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
6		ssequn1 3297	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
7		pweq 3569	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵)
8		eqimss 3201	. . . . . 6 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
9	7, 8	syl 14	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
10	6, 9	sylbi 120	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
11	5, 10	orim12i 754	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
12	11	orcoms 725	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
13		ssun 3306	. 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
14	12, 13	syl 14	1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

Syntax hints:   → wi 4   ∨ wo 703   = wceq 1348   ∪ cun 3119   ⊆ wss 3121  𝒫 cpw 3566

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 3568

This theorem is referenced by:  pwunim  4271