Theorem sspw 4510

Description: The powerclass preserves inclusion. See sspwb 5307 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5307 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 3922	. . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
2	1	com12 32	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
3		velpw 4502	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4		velpw 4502	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	2, 3, 4	3imtr4g 299	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
6	5	ssrdv 3921	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3881  𝒫 cpw 4497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499

This theorem is referenced by:  sspwi  4511  sspwd  4512  sspwb  5307  r1pwss  9197  elsigagen2  31517  measres  31591