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Theorem latasymd 14163
Description: Deduce equality from lattice ordering. (eqssd 3196 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 14160 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
93, 4, 5, 8syl3anc 1182 . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151
This theorem is referenced by:  latjidm  14180  latmidm  14192  latjass  14201  oldmm1  29407  olj01  29415  olm01  29426  cvlcvr1  29529  llnmlplnN  29728  2llnjaN  29755  2lplnja  29808  cdlema1N  29980  hlmod1i  30045  lautj  30282  lautm  30283  cdleme19a  30492  cdleme28b  30560  trljco  30929  dochvalr  31547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-poset 14080  df-lat 14152
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