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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 28957 | . . 3 ⊢ 0ℋ = {0ℎ} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ}) |
3 | ax-hv0cl 28707 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
4 | 3 | elexi 3511 | . . 3 ⊢ 0ℎ ∈ V |
5 | 4 | elsn2 4594 | . 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ) |
6 | 2, 5 | bitri 276 | 1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 ℋchba 28623 0ℎc0v 28628 0ℋc0h 28639 |
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-hv0cl 28707 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-v 3494 df-sn 4558 df-ch0 28957 |
This theorem is referenced by: ocin 29000 ocnel 29002 shuni 29004 choc0 29030 choc1 29031 omlsilem 29106 pjoc1i 29135 shne0i 29152 h1dn0 29256 spansnm0i 29354 nonbooli 29355 eleigvec 29661 cdjreui 30136 cdj3lem1 30138 |
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