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| Description: An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| atcv0 | ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ela 32358 | . 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 Cℋ cch 30948 0ℋc0h 30954 ⋖ℋ ccv 30983 HAtomscat 30984 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-at 32357 | 
| This theorem is referenced by: atcveq0 32367 atcv0eq 32398 | 
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