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Theorem shex 28378
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 28165 . . 3 ℋ ∈ V
21pwex 4997 . 2 𝒫 ℋ ∈ V
3 shss 28376 . . . 4 (𝑥S𝑥 ⊆ ℋ)
4 selpw 4309 . . . 4 (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
53, 4sylibr 224 . . 3 (𝑥S𝑥 ∈ 𝒫 ℋ)
65ssriv 3748 . 2 S ⊆ 𝒫 ℋ
72, 6ssexi 4955 1 S ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302  chil 28085   S csh 28094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-pow 4992  ax-hilex 28165
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-sh 28373
This theorem is referenced by:  chex  28392
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