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

Theorem elatcv0 29540
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 29538 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21baib 525 1 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145   class class class wbr 4787   C cch 28126  0c0h 28132   ccv 28161  HAtomscat 28162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-at 29537
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator