Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalemddea Structured version   Unicode version

Theorem dalemddea 30481
Description: Lemma for dath 30533. 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 982 . 2  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
d  e.  A )
31, 2sylbi 188 1  |-  ( ps 
->  d  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2599   class class class wbr 4212  (class class class)co 6081
This theorem is referenced by:  dalemswapyzps  30487  dalemrotps  30488  dalemcjden  30489  dalem21  30491  dalem23  30493  dalem24  30494  dalem25  30495  dalem27  30496  dalem56  30525
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
  Copyright terms: Public domain W3C validator