Theorem sspwimpcf 44918

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. sspwimpcf 44918, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 44919, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sspwimpcf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3482	. . . . . 6 ⊢ 𝑥 ∈ V
2		id 22	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
3		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴)
4		elpwi 4612	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6		sstr2 4002	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
7	6	impcom 407	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
8	2, 5, 7	syl2an 596	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵)
9		elpwg 4608	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
10	9	biimpar 477	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ 𝒫 𝐵)
11	1, 8, 10	eel021old 44698	. . . . 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 1925	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
14		df-ss 3980	. . . 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 44570	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-pw 4607

This theorem is referenced by: (None)