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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 27240 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | pwex 4774 | . 2 ⊢ 𝒫 ℋ ∈ V |
3 | shss 27451 | . . . 4 ⊢ (𝑥 ∈ Sℋ → 𝑥 ⊆ ℋ) | |
4 | selpw 4115 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℋ ↔ 𝑥 ⊆ ℋ) | |
5 | 3, 4 | sylibr 223 | . . 3 ⊢ (𝑥 ∈ Sℋ → 𝑥 ∈ 𝒫 ℋ) |
6 | 5 | ssriv 3572 | . 2 ⊢ Sℋ ⊆ 𝒫 ℋ |
7 | 2, 6 | ssexi 4731 | 1 ⊢ Sℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ℋchil 27160 Sℋ csh 27169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-sh 27448 |
This theorem is referenced by: chex 27467 |
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