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Theorem atcvr0 35364
Description: An atom covers zero. (atcv0 29757 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 2826 . . 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 35362 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃)))
65simplbda 495 1 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166   class class class wbr 4874  cfv 6124  Basecbs 16223  0.cp0 17391  ccvr 35338  Atomscatm 35339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-ats 35343
This theorem is referenced by:  0ltat  35367  leatb  35368  atnle0  35385  atlen0  35386  atcmp  35387  atcvreq0  35390  atcvr0eq  35502  lnnat  35503  athgt  35532  ps-2  35554  lhp0lt  36079
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