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Theorem latasymd 14486
Description: Deduce equality from lattice ordering. (eqssd 3365 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 14483 . . 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 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Latclat 14474
This theorem is referenced by:  latjidm  14503  latmidm  14515  latjass  14524  oldmm1  30015  olj01  30023  olm01  30034  cvlcvr1  30137  llnmlplnN  30336  2llnjaN  30363  2lplnja  30416  cdlema1N  30588  hlmod1i  30653  lautj  30890  lautm  30891  cdleme19a  31100  cdleme28b  31168  trljco  31537  dochvalr  32155
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-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-poset 14403  df-lat 14475
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