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Theorem atcvr0 35097
Description: An atom covers zero. (atcv0 29541 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 2771 . . 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 35095 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃)))
65simplbda 487 1 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145   class class class wbr 4787  cfv 6030  Basecbs 16064  0.cp0 17245  ccvr 35071  Atomscatm 35072
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  ax-sep 4916  ax-nul 4924  ax-pr 5035
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-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  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-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ats 35076
This theorem is referenced by:  0ltat  35100  leatb  35101  atnle0  35118  atlen0  35119  atcmp  35120  atcvreq0  35123  atcvr0eq  35235  lnnat  35236  athgt  35265  ps-2  35287  lhp0lt  35812
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