Theorem shex 31232

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 31019	. . 3 ⊢ ℋ ∈ V
2	1	pwex 5379	. 2 ⊢ 𝒫 ℋ ∈ V
3		shss 31230	. . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ)
4		velpw 4604	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
5	3, 4	sylibr 234	. . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ)
6	5	ssriv 3986	. 2 ⊢ Sℋ ⊆ 𝒫 ℋ
7	2, 6	ssexi 5321	1 ⊢ Sℋ ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950  𝒫 cpw 4599   ℋchba 30939   S_ℋ csh 30948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-pow 5364  ax-hilex 31019

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-sh 31227

This theorem is referenced by:  chex  31246