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Theorem shex 29475
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 29262 . . 3 ℋ ∈ V
21pwex 5298 . 2 𝒫 ℋ ∈ V
3 shss 29473 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 velpw 4535 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 233 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3921 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 5241 1 S ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3422  wss 3883  𝒫 cpw 4530  chba 29182   S csh 29191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-pow 5283  ax-hilex 29262
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-sh 29470
This theorem is referenced by:  chex  29489
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