Theorem elat 5738

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15.

Assertion

Ref	Expression
elat	⊢ (A ∈ Atoms ↔ (A ∈ Cℋ ∧ 0ℋ ⋖ A))

Proof of Theorem elat

Step	Hyp	Ref	Expression
1		df-at 5737	. . 3 ⊢ Atoms = {x ∈ Cℋ ∣0ℋ ⋖ x}
2	1	eleq2i 1153	. 2 ⊢ (A ∈ Atoms ↔ A ∈ {x ∈ Cℋ ∣0ℋ ⋖ x})
3		breq2 2066	. . 3 ⊢ (x = A → (0ℋ ⋖ x ↔ 0ℋ ⋖ A))
4	3	elrab 1422	. 2 ⊢ (A ∈ {x ∈ Cℋ ∣0ℋ ⋖ x} ↔ (A ∈ Cℋ ∧ 0ℋ ⋖ A))
5	2, 4	bitr 151	1 ⊢ (A ∈ Atoms ↔ (A ∈ Cℋ ∧ 0ℋ ⋖ A))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  {crab 1204   class class class wbr 2054   C_ℋ cch 4968  0_ℋc0h 4974  Atomscat 4980   ⋖ ccv 4981

This theorem is referenced by:  elat2 5739  atcv0 5740

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-at 5737

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