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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 28165 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | pwex 4997 | . 2 ⊢ 𝒫 ℋ ∈ V |
3 | shss 28376 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
4 | selpw 4309 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
5 | 3, 4 | sylibr 224 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) |
6 | 5 | ssriv 3748 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ |
7 | 2, 6 | ssexi 4955 | 1 ⊢ Sℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 𝒫 cpw 4302 ℋchil 28085 Sℋ csh 28094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-pow 4992 ax-hilex 28165 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-xp 5272 df-cnv 5274 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-sh 28373 |
This theorem is referenced by: chex 28392 |
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