Theorem sspwimp 44949

Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. For the biconditional, see sspwb 5390. The proof sspwimp 44949, using conventional notation, was translated from virtual deduction form, sspwimpVD 44950, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sspwimp	⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑥 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴)
5		elpwi 4557	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
7		sstr 3943	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
8	7	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
9	3, 6, 8	syl2an 596	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵)
10	2, 9	elpwgded 44596	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
11	2, 9, 10	uun0.1 44809	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
12	11	ex 412	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
13	12	alrimiv 1928	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
14		df-ss 3919	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
15	14	biimpri 228	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
16	13, 15	syl 17	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
17	16	iin1 44604	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  ⊤wtru 1542   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3902  𝒫 cpw 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-pw 4552

This theorem is referenced by: (None)