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Theorem latasymd 14373
Description: Deduce equality from lattice ordering. (eqssd 3282 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 14370 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
93, 4, 5, 8syl3anc 1183 . 2  |-  ( ph  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
101, 2, 9mpbi2and 887 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358   Basecbs 13356   lecple 13423   Latclat 14361
This theorem is referenced by:  latjidm  14390  latmidm  14402  latjass  14411  oldmm1  29478  olj01  29486  olm01  29497  cvlcvr1  29600  llnmlplnN  29799  2llnjaN  29826  2lplnja  29879  cdlema1N  30051  hlmod1i  30116  lautj  30353  lautm  30354  cdleme19a  30563  cdleme28b  30631  trljco  31000  dochvalr  31618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-nul 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984  df-poset 14290  df-lat 14362
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