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Theorem shex 28916
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 S ∈ V

Proof of Theorem shex
StepHypRef Expression
1 ax-hilex 28703 . . 3 ℋ ∈ V
21pwex 5272 . 2 𝒫 ℋ ∈ V
3 shss 28914 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 velpw 4543 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 235 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3968 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 5217 1 S ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535  chba 28623   S csh 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-pow 5257  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sh 28911
This theorem is referenced by:  chex  28930
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