Theorem sspw 4573

Description: The powerclass preserves inclusion. See sspwb 5430 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5430 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)

Assertion

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

Proof of Theorem sspw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3944	. . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
2	1	com12 33	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
3		velpw 4567	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4		velpw 4567	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	2, 3, 4	3imtr4g 299	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
6	5	ssrdv 3943	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2143   ⊆ wss 3905  𝒫 cpw 4562

This proof depends on 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 proof 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 3922  df-pw 4564

This theorem is used by:  sspwi  4574  sspwd  4575  sspwb  5430  r1pwss  9752  elsigagen2  34547  measres  34621