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Theorem dalemcceb 32966
Description: Lemma for dath 33013. 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 32960 . 2  |-  ( ps 
->  c  e.  A
)
3 eqid 2429 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 da.a1 . . 3  |-  A  =  ( Atoms `  K )
53, 4atbase 32567 . 2  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
62, 5syl 17 1  |-  ( ps 
->  c  e.  ( Base `  K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   Atomscatm 32541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ats 32545
This theorem is referenced by:  dalem21  32971  dalem25  32975  dalem38  32987  dalem39  32988  dalem44  32993  dalem45  32994  dalem48  32997  dalem52  33001
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