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Theorem pleval2 14414
Description: Less-than-or-equal in terms of less-than. (sspss 3438 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 14413 . . 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 14410 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
71, 2posref 14400 . . . . 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 4208 . . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   ltcplt 14390
This theorem is referenced by:  pltletr  14420  plelttr  14421  tosso  14457  ofldsqr  24232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-poset 14395  df-plt 14407
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