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Theorem pltval3 16206
Description: Alternate expression for less-than relation. (dfpss3 3518 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
pleval2.b  |-  B  =  ( Base `  K
)
pleval2.l  |-  .<_  =  ( le `  K )
pleval2.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltval3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )

Proof of Theorem pltval3
StepHypRef Expression
1 pleval2.l . . 3  |-  .<_  =  ( le `  K )
2 pleval2.s . . 3  |-  .<  =  ( lt `  K )
31, 2pltval 16199 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
4 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
54, 1posref 16189 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
653adant3 1027 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
7 breq1 4404 . . . . . . 7  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
86, 7syl5ibcom 224 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
98adantr 467 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  ->  Y  .<_  X ) )
104, 1posasymb 16191 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )
1110biimpd 211 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
1211expdimp 439 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( Y  .<_  X  ->  X  =  Y ) )
139, 12impbid 194 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  <->  Y  .<_  X ) )
1413necon3abid 2659 . . 3  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =/= 
Y  <->  -.  Y  .<_  X ) )
1514pm5.32da 646 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  X  =/=  Y )  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
163, 15bitrd 257 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   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-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  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:  tltnle  28416  opltcon3b  32764
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