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Theorem shex 28984
 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 28771 . . 3 ℋ ∈ V
21pwex 5262 . 2 𝒫 ℋ ∈ V
3 shss 28982 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 velpw 4525 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 237 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3955 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 5207 1 S ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  Vcvv 3479   ⊆ wss 3918  𝒫 cpw 4520   ℋchba 28691   Sℋ csh 28700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-pow 5247  ax-hilex 28771 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-sh 28979 This theorem is referenced by:  chex  28998
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