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

Theorem cdlemesner 30778
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
cdlemesner.l  |-  .<_  =  ( le `  K )
cdlemesner.j  |-  .\/  =  ( join `  K )
cdlemesner.a  |-  A  =  ( Atoms `  K )
cdlemesner.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemesner  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  =/=  R )

Proof of Theorem cdlemesner
StepHypRef Expression
1 nbrne2 4190 . . 3  |-  ( ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  R  =/=  S )
213ad2ant3 980 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  =/=  S )
32necomd 2650 1  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  =/=  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemeda  30780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173
  Copyright terms: Public domain W3C validator