Theorem shex

Description: The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shex	⊢ Sℋ ∈ V

Proof

Step	Hyp	Ref	Expression
1		ax-hilex	⊢ ℋ ∈ V
2	1	pwex	⊢ 𝒫 ℋ ∈ V
3		shss	⊢ ( 𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ )
4		velpw	⊢ ( 𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ )
5	3 4	sylibr	⊢ ( 𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ )
6	5	ssriv	⊢ Sℋ ⊆ 𝒫 ℋ
7	2 6	ssexi	⊢ Sℋ ∈ V