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Theorem atcvr0 33593
 Description: An atom covers zero. (atcv0 28585 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atomcvr0.z 0 = (0.‘𝐾)
atomcvr0.c 𝐶 = ( ⋖ ‘𝐾)
atomcvr0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atcvr0 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)

Proof of Theorem atcvr0
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 atomcvr0.z . . 3 0 = (0.‘𝐾)
3 atomcvr0.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 atomcvr0.a . . 3 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 33591 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃)))
65simplbda 652 1 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  0.cp0 16860   ⋖ ccvr 33567  Atomscatm 33568 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ats 33572 This theorem is referenced by:  0ltat  33596  leatb  33597  atnle0  33614  atlen0  33615  atcmp  33616  atcvreq0  33619  atcvr0eq  33730  lnnat  33731  athgt  33760  ps-2  33782  lhp0lt  34307
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