Theorem chex 31157

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 31143	. 2 ⊢ Sℋ ∈ V
2		chsssh 31156	. 2 ⊢ Cℋ ⊆ Sℋ
3	1, 2	ssexi 5257	1 ⊢ Cℋ ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3433   S_ℋ csh 30859   C_ℋ cch 30860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5231  ax-pow 5300  ax-hilex 30930

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-xp 5619  df-cnv 5621  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fv 6484  df-ov 7343  df-sh 31138  df-ch 31152

This theorem is referenced by:  isst  32144  ishst  32145