Theorem ela 23799

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
ela	|- ( A e. HAtoms <-> ( A e. CH /\ 0H <oH A ) )

Proof of Theorem ela

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4180	. 2 |- ( x = A -> ( 0H <oH x <-> 0H <oH A ) )
2		df-at 23798	. 2 |- HAtoms = { x e. CH | 0H <oH x }
3	1, 2	elrab2 3058	1 |- ( A e. HAtoms <-> ( A e. CH /\ 0H <oH A ) )

Syntax hints:   <-> wb 177   /\ wa 359   e. wcel 1721   class class class wbr 4176   CHcch 22389   0Hc0h 22395   <oH ccv 22424  HAtomscat 22425

This theorem is referenced by:  elat2  23800  elatcv0  23801  atcv0  23802

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-at 23798