Theorem shex 31244

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

Assertion

Ref	Expression
shex	⊢ Sℋ ∈ V

Proof of Theorem shex

Step	Hyp	Ref	Expression
1		ax-hilex 31031	. . 3 ⊢ ℋ ∈ V
2	1	pwex 5398	. 2 ⊢ 𝒫 ℋ ∈ V
3		shss 31242	. . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ)
4		velpw 4627	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
5	3, 4	sylibr 234	. . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ)
6	5	ssriv 4012	. 2 ⊢ Sℋ ⊆ 𝒫 ℋ
7	2, 6	ssexi 5340	1 ⊢ Sℋ ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   ℋchba 30951   S_ℋ csh 30960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pow 5383  ax-hilex 31031

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-sh 31239

This theorem is referenced by:  chex  31258