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Theorem shex 26841
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 26628 . . 3  |-  ~H  e.  _V
21pwex 4600 . 2  |-  ~P ~H  e.  _V
3 shss 26839 . . . 4  |-  ( x  e.  SH  ->  x  C_ 
~H )
4 selpw 3983 . . . 4  |-  ( x  e.  ~P ~H  <->  x  C_  ~H )
53, 4sylibr 215 . . 3  |-  ( x  e.  SH  ->  x  e.  ~P ~H )
65ssriv 3465 . 2  |-  SH  C_  ~P ~H
72, 6ssexi 4562 1  |-  SH  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1867   _Vcvv 3078    C_ wss 3433   ~Pcpw 3976   ~Hchil 26548   SHcsh 26557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-pow 4595  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-sh 26836
This theorem is referenced by:  chex  26855
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