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Theorem isat 30084
Description: The predicate "is an atom". (ela 23842 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 30083 . . 3  |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x }
)
65eleq2d 2503 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  P  e.  { x  e.  B  |  .0.  C x } ) )
7 breq2 4216 . . 3  |-  ( x  =  P  ->  (  .0.  C x  <->  .0.  C P ) )
87elrab 3092 . 2  |-  ( P  e.  { x  e.  B  |  .0.  C x }  <->  ( P  e.  B  /\  .0.  C P ) )
96, 8syl6bb 253 1  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  B  /\  .0.  C P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   class class class wbr 4212   ` cfv 5454   Basecbs 13469   0.cp0 14466    <o ccvr 30060   Atomscatm 30061
This theorem is referenced by:  isat2  30085  atcvr0  30086  atbase  30087  isat3  30105  1cvrco  30269  1cvrjat  30272  ltrnatb  30934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ats 30065
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