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Theorem latasym 14489
Description: A lattice ordering is asymetric. (eqss 3365 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
latref.b  |-  B  =  ( Base `  K
)
latref.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
latasym  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y ) )

Proof of Theorem latasym
StepHypRef Expression
1 latref.b . . 3  |-  B  =  ( Base `  K
)
2 latref.l . . 3  |-  .<_  =  ( le `  K )
31, 2latasymb 14488 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
43biimpd 200 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Latclat 14479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-poset 14408  df-lat 14480
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