MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latasymd Unicode version

Theorem latasymd 14441
Description: Deduce equality from lattice ordering. (eqssd 3325 analog.) (Contributed by NM, 18-Nov-2011.)
Hypotheses
Ref Expression
latasymd.b  |-  B  =  ( Base `  K
)
latasymd.l  |-  .<_  =  ( le `  K )
latasymd.3  |-  ( ph  ->  K  e.  Lat )
latasymd.4  |-  ( ph  ->  X  e.  B )
latasymd.5  |-  ( ph  ->  Y  e.  B )
latasymd.6  |-  ( ph  ->  X  .<_  Y )
latasymd.7  |-  ( ph  ->  Y  .<_  X )
Assertion
Ref Expression
latasymd  |-  ( ph  ->  X  =  Y )

Proof of Theorem latasymd
StepHypRef Expression
1 latasymd.6 . 2  |-  ( ph  ->  X  .<_  Y )
2 latasymd.7 . 2  |-  ( ph  ->  Y  .<_  X )
3 latasymd.3 . . 3  |-  ( ph  ->  K  e.  Lat )
4 latasymd.4 . . 3  |-  ( ph  ->  X  e.  B )
5 latasymd.5 . . 3  |-  ( ph  ->  Y  e.  B )
6 latasymd.b . . . 4  |-  B  =  ( Base `  K
)
7 latasymd.l . . . 4  |-  .<_  =  ( le `  K )
86, 7latasymb 14438 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
93, 4, 5, 8syl3anc 1184 . 2  |-  ( ph  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
101, 2, 9mpbi2and 888 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Latclat 14429
This theorem is referenced by:  latjidm  14458  latmidm  14470  latjass  14479  oldmm1  29700  olj01  29708  olm01  29719  cvlcvr1  29822  llnmlplnN  30021  2llnjaN  30048  2lplnja  30101  cdlema1N  30273  hlmod1i  30338  lautj  30575  lautm  30576  cdleme19a  30785  cdleme28b  30853  trljco  31222  dochvalr  31840
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  ax-nul 4298
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-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  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-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-poset 14358  df-lat 14430
  Copyright terms: Public domain W3C validator