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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 31083 | . . 3 ⊢ 0ℋ = {0ℎ} | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ}) |
3 | ax-hv0cl 30833 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
4 | 3 | elexi 3493 | . . 3 ⊢ 0ℎ ∈ V |
5 | 4 | elsn2 4672 | . 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ) |
6 | 2, 5 | bitri 274 | 1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 {csn 4632 ℋchba 30749 0ℎc0v 30754 0ℋc0h 30765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-hv0cl 30833 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-sn 4633 df-ch0 31083 |
This theorem is referenced by: ocin 31126 ocnel 31128 shuni 31130 choc0 31156 choc1 31157 omlsilem 31232 pjoc1i 31261 shne0i 31278 h1dn0 31382 spansnm0i 31480 nonbooli 31481 eleigvec 31787 cdjreui 32262 cdj3lem1 32264 |
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