Theorem shex 31241

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 31028	. . 3 ⊢ ℋ ∈ V
2	1	pwex 5386	. 2 ⊢ 𝒫 ℋ ∈ V
3		shss 31239	. . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ)
4		velpw 4610	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ)
5	3, 4	sylibr 234	. . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ)
6	5	ssriv 3999	. 2 ⊢ Sℋ ⊆ 𝒫 ℋ
7	2, 6	ssexi 5328	1 ⊢ Sℋ ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605   ℋchba 30948   S_ℋ csh 30957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-sh 31236

This theorem is referenced by:  chex  31255