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Theorem pleval2 14351
Description: Less-than-or-equal in terms of less-than. (sspss 3391 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pleval2.b  |-  B  =  ( Base `  K
)
pleval2.l  |-  .<_  =  ( le `  K )
pleval2.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pleval2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )

Proof of Theorem pleval2
StepHypRef Expression
1 pleval2.b . . . 4  |-  B  =  ( Base `  K
)
2 pleval2.l . . . 4  |-  .<_  =  ( le `  K )
3 pleval2.s . . . 4  |-  .<  =  ( lt `  K )
41, 2, 3pleval2i 14350 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
543adant1 975 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( X  .<  Y  \/  X  =  Y ) ) )
62, 3pltle 14347 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
71, 2posref 14337 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
873adant3 977 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
9 breq2 4159 . . . 4  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
108, 9syl5ibcom 212 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
116, 10jaod 370 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<  Y  \/  X  =  Y )  ->  X  .<_  Y )
)
125, 11impbid 184 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   Posetcpo 14326   ltcplt 14327
This theorem is referenced by:  pltletr  14357  plelttr  14358  tosso  14394  ofldsqr  24068
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-poset 14332  df-plt 14344
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