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Theorem shex 28197
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 27984 . . 3 ℋ ∈ V
21pwex 4878 . 2 𝒫 ℋ ∈ V
3 shss 28195 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 selpw 4198 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 224 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3640 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 4836 1 S ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191  chil 27904   S csh 27913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-sh 28192
This theorem is referenced by:  chex  28211
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