Theorem pwssunim 4243

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 3280	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
2		pweq 3546	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴)
3		eqimss 3182	. . . . . 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 3277	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
7		pweq 3546	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵)
8		eqimss 3182	. . . . . 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 749	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
12	11	orcoms 720	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
13		ssun 3286	. 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 698   = wceq 1335   ∪ cun 3100   ⊆ wss 3102  𝒫 cpw 3543

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545

This theorem is referenced by:  pwunim  4245