Theorem chex 29877

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 29863	. 2 ⊢ Sℋ ∈ V
2		chsssh 29876	. 2 ⊢ Cℋ ⊆ Sℋ
3	1, 2	ssexi 5267	1 ⊢ Cℋ ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3441   S_ℋ csh 29579   C_ℋ cch 29580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5244  ax-pow 5309  ax-hilex 29650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-br 5094  df-opab 5156  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fv 6488  df-ov 7341  df-sh 29858  df-ch 29872

This theorem is referenced by:  isst  30864  ishst  30865