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Theorem shex 22707
Description: The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shex  |-  SH  e.  _V

Proof of Theorem shex
StepHypRef Expression
1 ax-hilex 22495 . . 3  |-  ~H  e.  _V
21pwex 4375 . 2  |-  ~P ~H  e.  _V
3 shss 22705 . . . 4  |-  ( x  e.  SH  ->  x  C_ 
~H )
4 vex 2952 . . . . 5  |-  x  e. 
_V
54elpw 3798 . . . 4  |-  ( x  e.  ~P ~H  <->  x  C_  ~H )
63, 5sylibr 204 . . 3  |-  ( x  e.  SH  ->  x  e.  ~P ~H )
76ssriv 3345 . 2  |-  SH  C_  ~P ~H
82, 7ssexi 4341 1  |-  SH  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2949    C_ wss 3313   ~Pcpw 3792   ~Hchil 22415   SHcsh 22424
This theorem is referenced by:  chex  22722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-pow 4370  ax-hilex 22495
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-cnv 4879  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-sh 22702
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