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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 28661 | . . 3 ⊢ 0ℋ = {0ℎ} | |
2 | 1 | eleq2i 2898 | . 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ}) |
3 | ax-hv0cl 28411 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
4 | 3 | elexi 3430 | . . 3 ⊢ 0ℎ ∈ V |
5 | 4 | elsn2 4434 | . 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ) |
6 | 2, 5 | bitri 267 | 1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1656 ∈ wcel 2164 {csn 4399 ℋchba 28327 0ℎc0v 28332 0ℋc0h 28343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 ax-hv0cl 28411 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-sn 4400 df-ch0 28661 |
This theorem is referenced by: ocin 28706 ocnel 28708 shuni 28710 choc0 28736 choc1 28737 omlsilem 28812 pjoc1i 28841 shne0i 28858 h1dn0 28962 spansnm0i 29060 nonbooli 29061 eleigvec 29367 cdjreui 29842 cdj3lem1 29844 |
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