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	⊢ Sℋ ∈ V

Proof of Theorem shex

Step	Hyp	Ref	Expression
1		ax-hilex 28165	. . 3 ⊢ ℋ ∈ V
2	1	pwex 4997	. 2 ⊢ 𝒫 ℋ ∈ V
3		shss 28376	. . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ)
4		selpw 4309	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
5	3, 4	sylibr 224	. . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ)
6	5	ssriv 3748	. 2 ⊢ Sℋ ⊆ 𝒫 ℋ
7	2, 6	ssexi 4955	1 ⊢ Sℋ ∈ V

Syntax hints:   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715  𝒫 cpw 4302   ℋchil 28085   S_ℋ 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