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Theorem isat2 30159
Description: The predicate "is an atom". (elatcv0 23849 analog.) (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
isatom.b  |-  B  =  ( Base `  K
)
isatom.z  |-  .0.  =  ( 0. `  K )
isatom.c  |-  C  =  (  <o  `  K )
isatom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
isat2  |-  ( ( K  e.  D  /\  P  e.  B )  ->  ( P  e.  A  <->  .0. 
C P ) )

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  =  (  <o  `  K )
4 isatom.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 30158 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  C P ) ) )
65baibd 877 1  |-  ( ( K  e.  D  /\  P  e.  B )  ->  ( P  e.  A  <->  .0. 
C P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457   Basecbs 13474   0.cp0 14471    <o ccvr 30134   Atomscatm 30135
This theorem is referenced by:  llncvrlpln  30429  lplncvrlvol  30487  lhpm0atN  30900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ats 30139
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