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| Description: The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| shex | ⊢ Sℋ ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-hilex 31019 | . . 3 ⊢ ℋ ∈ V | |
| 2 | 1 | pwex 5379 | . 2 ⊢ 𝒫 ℋ ∈ V | 
| 3 | shss 31230 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
| 4 | velpw 4604 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
| 5 | 3, 4 | sylibr 234 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) | 
| 6 | 5 | ssriv 3986 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ | 
| 7 | 2, 6 | ssexi 5321 | 1 ⊢ Sℋ ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 ℋchba 30939 Sℋ csh 30948 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-hilex 31019 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-sh 31227 | 
| This theorem is referenced by: chex 31246 | 
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