Theorem sspwimp 45606

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 5432. The proof sspwimp 45606, using conventional notation, was translated from virtual deduction form, sspwimpVD 45607, 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 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑥 ∈ V)
3		id 23	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4		id 23	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴)
5		elpwi 4570	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6	4, 5	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
7		sstr 3946	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
8	7	ancoms 463	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
9	3, 6, 8	syl2an 607	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵)
10	2, 9	elpwgded 45253	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
11	2, 9, 10	uun0.1 45466	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
12	11	ex 417	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
13	12	alrimiv 1957	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
14		df-ss 3923	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
15	14	biimpri 231	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
16	13, 15	syl 18	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
17	16	iin1 45261	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568  ⊤wtru 1571   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  𝒫 cpw 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-pw 4565

This theorem is referenced by: (None)