Theorem shex 28197

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 27984	. . 3 ⊢ ℋ ∈ V
2	1	pwex 4878	. 2 ⊢ 𝒫 ℋ ∈ V
3		shss 28195	. . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ)
4		selpw 4198	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
5	3, 4	sylibr 224	. . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ)
6	5	ssriv 3640	. 2 ⊢ Sℋ ⊆ 𝒫 ℋ
7	2, 6	ssexi 4836	1 ⊢ Sℋ ∈ V

Syntax hints:   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191   ℋchil 27904   S_ℋ csh 27913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-hilex 27984

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-sh 28192

This theorem is referenced by:  chex  28211