Theorem sspwimp

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 . The proof sspwimp , using conventional notation, was translated from virtual deduction form, sspwimpVD , using a translation program. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	a1i	⊢ ( ⊤ → 𝑥 ∈ V )
3		id	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵 )
4		id	⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐴 )
5		elpwi	⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴 )
6	4 5	syl	⊢ ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴 )
7		sstr	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → 𝑥 ⊆ 𝐵 )
8	7	ancoms	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ) → 𝑥 ⊆ 𝐵 )
9	3 6 8	syl2an	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) → 𝑥 ⊆ 𝐵 )
10	2 9	elpwgded	⊢ ( ( ⊤ ∧ ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) ) → 𝑥 ∈ 𝒫 𝐵 )
11	2 9 10	uun0.1	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ) → 𝑥 ∈ 𝒫 𝐵 )
12	11	ex	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) )
13	12	alrimiv	⊢ ( 𝐴 ⊆ 𝐵 → ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) )
14		df-ss	⊢ ( 𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) )
15	14	biimpri	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵 ) → 𝒫 𝐴 ⊆ 𝒫 𝐵 )
16	13 15	syl	⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 )
17	16	iin1	⊢ ( 𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 )