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Theorem dalemcceb 33299
Description: Lemma for dath 33346. 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 33293 . 2  |-  ( ps 
->  c  e.  A
)
3 eqid 2462 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 da.a1 . . 3  |-  A  =  ( Atoms `  K )
53, 4atbase 32900 . 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   Atomscatm 32874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ats 32878
This theorem is referenced by:  dalem21  33304  dalem25  33308  dalem38  33320  dalem39  33321  dalem44  33326  dalem45  33327  dalem48  33330  dalem52  33334
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