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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 28782 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5246 | . 2 ⊢ 𝒫 ℋ ∈ V |
3 | shss 28993 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
4 | velpw 4502 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
5 | 3, 4 | sylibr 237 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) |
6 | 5 | ssriv 3919 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ |
7 | 2, 6 | ssexi 5190 | 1 ⊢ Sℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ℋchba 28702 Sℋ csh 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-pow 5231 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sh 28990 |
This theorem is referenced by: chex 29009 |
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