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Theorem dalemccea 29872
Description: Lemma for dath 29925. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypothesis
Ref Expression
da.ps0  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
Assertion
Ref Expression
dalemccea  |-  ( ps 
->  c  e.  A
)

Proof of Theorem dalemccea
StepHypRef Expression
1 da.ps0 . 2  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
2 simp1l 979 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
c  e.  A )
31, 2sylbi 187 1  |-  ( ps 
->  c  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  dalemcceb  29878  dalemswapyzps  29879  dalemrotps  29880  dalemcjden  29881  dalem23  29885  dalem24  29886  dalem25  29887  dalem27  29888  dalem28  29889  dalem38  29899  dalem39  29900  dalem44  29905  dalem51  29912  dalem56  29917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
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