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Mirrors > Home > HSE Home > Th. List > elatcv0 | Structured version Visualization version GIF version |
Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elatcv0 | ⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ela 29754 | . 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) | |
2 | 1 | baib 533 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2166 class class class wbr 4874 Cℋ cch 28342 0ℋc0h 28348 ⋖ℋ ccv 28377 HAtomscat 28378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-at 29753 |
This theorem is referenced by: (None) |
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