Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isat2 Unicode version

Theorem isat2 29782
Description: The predicate "is an atom". (elatcv0 23805 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 29781 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  C P ) ) )
65baibd 876 1  |-  ( ( K  e.  D  /\  P  e.  B )  ->  ( P  e.  A  <->  .0. 
C P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421   Basecbs 13432   0.cp0 14429    <o ccvr 29757   Atomscatm 29758
This theorem is referenced by:  llncvrlpln  30052  lplncvrlvol  30110  lhpm0atN  30523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ats 29762
  Copyright terms: Public domain W3C validator