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Theorem dalemddea 29246
Description: Lemma for dath 29298. 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
dalemddea  |-  ( ps 
->  d  e.  A
)

Proof of Theorem dalemddea
StepHypRef Expression
1 da.ps0 . 2  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
2 simp1r 980 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
d  e.  A )
31, 2sylbi 187 1  |-  ( ps 
->  d  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  dalemswapyzps  29252  dalemrotps  29253  dalemcjden  29254  dalem21  29256  dalem23  29258  dalem24  29259  dalem25  29260  dalem27  29261  dalem56  29290
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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