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Metamath Proof Explorer

Theorem sspw

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

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

Proof

Step	Hyp	Ref	Expression
1		sstr2	⊢ ( 𝑥 ⊆ 𝐴 → ( 𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵 ) )
2	1	com12	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) )
3		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 )
4		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵 )
5	2 3 4	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) )
6	5	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 )