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Theorem pleval2 16204
Description: Less-than-or-equal in terms of less-than. (sspss 3531 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 16203 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
543adant1 1025 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( X  .<  Y  \/  X  =  Y ) ) )
62, 3pltle 16200 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
71, 2posref 16189 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
873adant3 1027 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
9 breq2 4405 . . . 4  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
108, 9syl5ibcom 224 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
116, 10jaod 382 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<  Y  \/  X  =  Y )  ->  X  .<_  Y )
)
125, 11impbid 194 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ w3a 984    = wceq 1443    e. wcel 1886   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   Posetcpo 16178   ltcplt 16179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-preset 16166  df-poset 16184  df-plt 16197
This theorem is referenced by:  pltletr  16210  plelttr  16211  tosso  16275  tlt3  28419  orngsqr  28560
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