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Theorem shex 27918
 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 27705 . . 3 ℋ ∈ V
21pwex 4808 . 2 𝒫 ℋ ∈ V
3 shss 27916 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 selpw 4137 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 224 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3587 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 4763 1 S ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1987  Vcvv 3186   ⊆ wss 3555  𝒫 cpw 4130   ℋchil 27625   Sℋ csh 27634 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-pow 4803  ax-hilex 27705 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-sh 27913 This theorem is referenced by:  chex  27932
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