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Theorem dalemcceb 30486
Description: Lemma for dath 30533. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypotheses
Ref Expression
da.ps0  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
da.a1  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalemcceb  |-  ( ps 
->  c  e.  ( Base `  K ) )

Proof of Theorem dalemcceb
StepHypRef Expression
1 da.ps0 . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
21dalemccea 30480 . 2  |-  ( ps 
->  c  e.  A
)
3 eqid 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 da.a1 . . 3  |-  A  =  ( Atoms `  K )
53, 4atbase 30087 . 2  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
62, 5syl 16 1  |-  ( ps 
->  c  e.  ( Base `  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   Atomscatm 30061
This theorem is referenced by:  dalem21  30491  dalem25  30495  dalem38  30507  dalem39  30508  dalem44  30513  dalem45  30514  dalem48  30517  dalem52  30521
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-13 1727  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-pow 4377  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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