Theorem shsspwh

Description: Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsspwh	⊢ Sℋ ⊆ 𝒫 ℋ

Proof

Step	Hyp	Ref	Expression
1		pwuni	⊢ Sℋ ⊆ 𝒫 ∪ Sℋ
2		helsh	⊢ ℋ ∈ Sℋ
3		shss	⊢ ( 𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ )
4	3	rgen	⊢ ∀ 𝑥 ∈ Sℋ 𝑥 ⊆ ℋ
5		ssunieq	⊢ ( ( ℋ ∈ Sℋ ∧ ∀ 𝑥 ∈ Sℋ 𝑥 ⊆ ℋ ) → ℋ = ∪ Sℋ )
6	2 4 5	mp2an	⊢ ℋ = ∪ Sℋ
7	6	pweqi	⊢ 𝒫 ℋ = 𝒫 ∪ Sℋ
8	1 7	sseqtrri	⊢ Sℋ ⊆ 𝒫 ℋ