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Theorem dalem-ddly 33295
Description: Lemma for dath 33345. 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-ddly  |-  ( ps 
->  -.  d  .<_  Y )

Proof of Theorem dalem-ddly
StepHypRef Expression
1 da.ps0 . 2  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
2 simp32 1051 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  ->  -.  d  .<_  Y )
31, 2sylbi 200 1  |-  ( ps 
->  -.  d  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    e. wcel 1897    =/= wne 2632   class class class wbr 4415  (class class class)co 6314
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 190  df-an 377  df-3an 993
This theorem is referenced by:  dalemswapyzps  33299  dalemrotps  33300  dalem23  33305  dalem24  33306  dalem25  33307
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