Theorem chex 31030

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

Assertion

Ref	Expression
chex	⊢ Cℋ ∈ V

Proof of Theorem chex

Step	Hyp	Ref	Expression
1		shex 31016	. 2 ⊢ Sℋ ∈ V
2		chsssh 31029	. 2 ⊢ Cℋ ⊆ Sℋ
3	1, 2	ssexi 5317	1 ⊢ Cℋ ∈ V

Syntax hints:   ∈ wcel 2099  Vcvv 3470   S_ℋ csh 30732   C_ℋ cch 30733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-pow 5360  ax-hilex 30803

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fv 6551  df-ov 7418  df-sh 31011  df-ch 31025

This theorem is referenced by:  isst  32017  ishst  32018