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Theorem dalemccea 29797
Description: Lemma for dath 29850. 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 981 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
c  e.  A )
31, 2sylbi 188 1  |-  ( ps 
->  c  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    =/= wne 2550   class class class wbr 4153  (class class class)co 6020
This theorem is referenced by:  dalemcceb  29803  dalemswapyzps  29804  dalemrotps  29805  dalemcjden  29806  dalem23  29810  dalem24  29811  dalem25  29812  dalem27  29813  dalem28  29814  dalem38  29824  dalem39  29825  dalem44  29830  dalem51  29837  dalem56  29842
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 178  df-an 361  df-3an 938
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