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Theorem shex 22556
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 22344 . . 3  |-  ~H  e.  _V
21pwex 4317 . 2  |-  ~P ~H  e.  _V
3 shss 22554 . . . 4  |-  ( x  e.  SH  ->  x  C_ 
~H )
4 vex 2896 . . . . 5  |-  x  e. 
_V
54elpw 3742 . . . 4  |-  ( x  e.  ~P ~H  <->  x  C_  ~H )
63, 5sylibr 204 . . 3  |-  ( x  e.  SH  ->  x  e.  ~P ~H )
76ssriv 3289 . 2  |-  SH  C_  ~P ~H
82, 7ssexi 4283 1  |-  SH  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   _Vcvv 2893    C_ wss 3257   ~Pcpw 3736   ~Hchil 22264   SHcsh 22273
This theorem is referenced by:  chex  22571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-pow 4312  ax-hilex 22344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-br 4148  df-opab 4202  df-xp 4818  df-cnv 4820  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-sh 22551
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