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Theorem dalemccea 30494
Description: Lemma for dath 30547. 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 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874
This theorem is referenced by:  dalemcceb  30500  dalemswapyzps  30501  dalemrotps  30502  dalemcjden  30503  dalem23  30507  dalem24  30508  dalem25  30509  dalem27  30510  dalem28  30511  dalem38  30521  dalem39  30522  dalem44  30527  dalem51  30534  dalem56  30539
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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