HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  elatcv0 Structured version   Visualization version   GIF version

Theorem elatcv0 31325
Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
elatcv0 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))

Proof of Theorem elatcv0
StepHypRef Expression
1 ela 31323 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21baib 537 1 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107   class class class wbr 5106   C cch 29913  0c0h 29919   ccv 29948  HAtomscat 29949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-at 31322
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator