HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  atcv0 Structured version   Visualization version   GIF version

Theorem atcv0 32361
Description: An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
atcv0 (𝐴 ∈ HAtoms → 0 𝐴)

Proof of Theorem atcv0
StepHypRef Expression
1 ela 32358 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21simprbi 496 1 (𝐴 ∈ HAtoms → 0 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   class class class wbr 5143   C cch 30948  0c0h 30954   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
  Copyright terms: Public domain W3C validator