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Theorem dalem-ccly 30419
Description: Lemma for dath 30470. 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
dalem-ccly  |-  ( ps 
->  -.  c  .<_  Y )

Proof of Theorem dalem-ccly
StepHypRef Expression
1 da.ps0 . 2  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
21simp2bi 973 1  |-  ( ps 
->  -.  c  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073
This theorem is referenced by:  dalemswapyzps  30424  dalemrotps  30425  dalem21  30428  dalem23  30430  dalem24  30431  dalem39  30445  dalem48  30454
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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