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Theorem isat2 36461
Description: The predicate "is an atom". (elatcv0 30102 analog.) (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
isatom.b 𝐵 = (Base‘𝐾)
isatom.z 0 = (0.‘𝐾)
isatom.c 𝐶 = ( ⋖ ‘𝐾)
isatom.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
isat2 ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))

Proof of Theorem isat2
StepHypRef Expression
1 isatom.b . . 3 𝐵 = (Base‘𝐾)
2 isatom.z . . 3 0 = (0.‘𝐾)
3 isatom.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 isatom.a . . 3 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 36460 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
65baibd 543 1 ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115   class class class wbr 5039  cfv 6328  Basecbs 16461  0.cp0 17625  ccvr 36436  Atomscatm 36437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ats 36441
This theorem is referenced by:  llncvrlpln  36732  lplncvrlvol  36790  lhpm0atN  37203
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