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Theorem dalemccnedd 30421
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
dalemccnedd  |-  ( ps 
->  c  =/=  d
)

Proof of Theorem dalemccnedd
StepHypRef Expression
1 da.ps0 . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
2 simp31 993 . . 3  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
d  =/=  c )
31, 2sylbi 188 . 2  |-  ( ps 
->  d  =/=  c
)
43necomd 2681 1  |-  ( ps 
->  c  =/=  d
)
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  dalemcjden  30426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-cleq 2428  df-ne 2600
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