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| Mirrors > Home > HSE Home > Th. List > elch0 | Structured version Visualization version GIF version | ||
| Description: Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| elch0 | ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ch0 31272 | . . 3 ⊢ 0ℋ = {0ℎ} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ}) |
| 3 | ax-hv0cl 31022 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
| 4 | 3 | elexi 3503 | . . 3 ⊢ 0ℎ ∈ V |
| 5 | 4 | elsn2 4665 | . 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ) |
| 6 | 2, 5 | bitri 275 | 1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {csn 4626 ℋchba 30938 0ℎc0v 30943 0ℋc0h 30954 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-hv0cl 31022 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sn 4627 df-ch0 31272 |
| This theorem is referenced by: ocin 31315 ocnel 31317 shuni 31319 choc0 31345 choc1 31346 omlsilem 31421 pjoc1i 31450 shne0i 31467 h1dn0 31571 spansnm0i 31669 nonbooli 31670 eleigvec 31976 cdjreui 32451 cdj3lem1 32453 |
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