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Theorem cdlemesner 29174
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 3938 . . 3  |-  ( ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  R  =/=  S )
213ad2ant3 983 . 2  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  =/=  S )
32necomd 2495 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdlemeda  29176
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921
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