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Mirrors > Home > HSE Home > Th. List > shex | Structured version Visualization version GIF version |
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 29262 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5298 | . 2 ⊢ 𝒫 ℋ ∈ V |
3 | shss 29473 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
4 | velpw 4535 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
5 | 3, 4 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) |
6 | 5 | ssriv 3921 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ |
7 | 2, 6 | ssexi 5241 | 1 ⊢ Sℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 Vcvv 3422 ⊆ wss 3883 𝒫 cpw 4530 ℋchba 29182 Sℋ csh 29191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-pow 5283 ax-hilex 29262 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5586 df-cnv 5588 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-sh 29470 |
This theorem is referenced by: chex 29489 |
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