Theorem sspwimpcf 45519

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 45519, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 45520, 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 3467	. . . . . 6 ⊢ 𝑥 ∈ V
2		id 23	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
3		id 23	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴)
4		elpwi 4574	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
5	3, 4	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6		sstr2 3952	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
7	6	impcom 412	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
8	2, 5, 7	syl2an 607	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵)
9		elpwg 4570	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
10	9	biimpar 482	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ 𝒫 𝐵)
11	1, 8, 10	eel021old 45300	. . . . 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 1954	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
14		df-ss 3930	. . . 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 45172	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1565   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-pw 4569

This theorem is referenced by: (None)