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Theorem atcvr0 29403
Description: An atom covers zero. (atcv0 23693 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atomcvr0.z  |-  .0.  =  ( 0. `  K )
atomcvr0.c  |-  C  =  (  <o  `  K )
atomcvr0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvr0  |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )

Proof of Theorem atcvr0
StepHypRef Expression
1 eqid 2387 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 atomcvr0.z . . 3  |-  .0.  =  ( 0. `  K )
3 atomcvr0.c . . 3  |-  C  =  (  <o  `  K )
4 atomcvr0.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isat 29401 . 2  |-  ( K  e.  D  ->  ( P  e.  A  <->  ( P  e.  ( Base `  K
)  /\  .0.  C P ) ) )
65simplbda 608 1  |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394   Basecbs 13396   0.cp0 14393    <o ccvr 29377   Atomscatm 29378
This theorem is referenced by:  0ltat  29406  leatb  29407  atnle0  29424  atlen0  29425  atcmp  29426  atcvreq0  29429  atcvr0eq  29540  lnnat  29541  athgt  29570  ps-2  29592  lhp0lt  30117
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ats 29382
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