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Theorem shex 21791
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 21579 . . 3  |-  ~H  e.  _V
21pwex 4193 . 2  |-  ~P ~H  e.  _V
3 shss 21789 . . . 4  |-  ( x  e.  SH  ->  x  C_ 
~H )
4 vex 2791 . . . . 5  |-  x  e. 
_V
54elpw 3631 . . . 4  |-  ( x  e.  ~P ~H  <->  x  C_  ~H )
63, 5sylibr 203 . . 3  |-  ( x  e.  SH  ->  x  e.  ~P ~H )
76ssriv 3184 . 2  |-  SH  C_  ~P ~H
82, 7ssexi 4159 1  |-  SH  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ~Hchil 21499   SHcsh 21508
This theorem is referenced by:  chex  21806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-pow 4188  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sh 21786
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