Theorem chex 29567

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 29553	. 2 ⊢ Sℋ ∈ V
2		chsssh 29566	. 2 ⊢ Cℋ ⊆ Sℋ
3	1, 2	ssexi 5249	1 ⊢ Cℋ ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3430   S_ℋ csh 29269   C_ℋ cch 29270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-pow 5291  ax-hilex 29340

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-xp 5594  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fv 6438  df-ov 7271  df-sh 29548  df-ch 29562

This theorem is referenced by:  isst  30554  ishst  30555