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	$$⊢{\mathbf{S}}_{\mathscr{H}}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		ax-hilex	$$⊢\mathscr{H}\in \mathrm{V}$$
2	1	pwex	$$⊢𝒫\mathscr{H}\in \mathrm{V}$$
3		shss	$$⊢x\in {\mathbf{S}}_{\mathscr{H}}\to x\subseteq \mathscr{H}$$
4		velpw	$$⊢x\in 𝒫\mathscr{H}\leftrightarrow x\subseteq \mathscr{H}$$
5	3 4	sylibr	$$⊢x\in {\mathbf{S}}_{\mathscr{H}}\to x\in 𝒫\mathscr{H}$$
6	5	ssriv	$$⊢{\mathbf{S}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$
7	2 6	ssexi	$$⊢{\mathbf{S}}_{\mathscr{H}}\in \mathrm{V}$$