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Theorem isat 29476
Description: The predicate "is an atom". (ela 22919 analog.) (Contributed by NM, 18-Sep-2011.)
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
isat  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  C P ) ) )

Proof of Theorem isat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isatom.b . . . 4  |-  B  =  ( Base `  K
)
2 isatom.z . . . 4  |-  .0.  =  ( 0. `  K )
3 isatom.c . . . 4  |-  C  =  (  <o  `  K )
4 isatom.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4pats 29475 . . 3  |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x }
)
65eleq2d 2350 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  P  e.  { x  e.  B  |  .0.  C x } ) )
7 breq2 4027 . . 3  |-  ( x  =  P  ->  (  .0.  C x  <->  .0.  C P ) )
87elrab 2923 . 2  |-  ( P  e.  { x  e.  B  |  .0.  C x }  <->  ( P  e.  B  /\  .0.  C P ) )
96, 8syl6bb 252 1  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  C P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   ` cfv 5255   Basecbs 13148   0.cp0 14143    <o ccvr 29452   Atomscatm 29453
This theorem is referenced by:  isat2  29477  atcvr0  29478  atbase  29479  isat3  29497  1cvrco  29661  1cvrjat  29664  ltrnatb  30326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ats 29457
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