Theorem chex 31258

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 31244	. 2 ⊢ Sℋ ∈ V
2		chsssh 31257	. 2 ⊢ Cℋ ⊆ Sℋ
3	1, 2	ssexi 5340	1 ⊢ Cℋ ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488   S_ℋ csh 30960   C_ℋ cch 30961

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-uni 4932  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-iota 6525  df-fv 6581  df-ov 7451  df-sh 31239  df-ch 31253

This theorem is referenced by:  isst  32245  ishst  32246