Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28703 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5272 | . 2 ⊢ 𝒫 ℋ ∈ V |
3 | shss 28914 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
4 | velpw 4543 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
5 | 3, 4 | sylibr 235 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) |
6 | 5 | ssriv 3968 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ |
7 | 2, 6 | ssexi 5217 | 1 ⊢ Sℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ℋchba 28623 Sℋ csh 28632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-pow 5257 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-sh 28911 |
This theorem is referenced by: chex 28930 |
Copyright terms: Public domain | W3C validator |