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Theorem dalemccea 33635
Description: Lemma for dath 33688. 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 1012 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
c  e.  A )
31, 2sylbi 195 1  |-  ( ps 
->  c  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2644   class class class wbr 4392  (class class class)co 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967
This theorem is referenced by:  dalemcceb  33641  dalemswapyzps  33642  dalemrotps  33643  dalemcjden  33644  dalem23  33648  dalem24  33649  dalem25  33650  dalem27  33651  dalem28  33652  dalem38  33662  dalem39  33663  dalem44  33668  dalem51  33675  dalem56  33680
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