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Theorem pleval2 14099
Description: Less-than-or-equal in terms of less-than. (sspss 3275 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 14098 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
543adant1 973 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( X  .<  Y  \/  X  =  Y ) ) )
62, 3pltle 14095 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
71, 2posref 14085 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
873adant3 975 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
9 breq2 4027 . . . 4  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
108, 9syl5ibcom 211 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
116, 10jaod 369 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<  Y  \/  X  =  Y )  ->  X  .<_  Y )
)
125, 11impbid 183 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 176    \/ wo 357    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075
This theorem is referenced by:  pltletr  14105  plelttr  14106  tosso  14142
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-mo 2148  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-poset 14080  df-plt 14092
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