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Theorem tltnle 28102
Description: In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 15920. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypotheses
Ref Expression
tleile.b  |-  B  =  ( Base `  K
)
tleile.l  |-  .<_  =  ( le `  K )
tltnle.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
tltnle  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  -.  Y  .<_  X ) )

Proof of Theorem tltnle
StepHypRef Expression
1 tospos 28098 . . 3  |-  ( K  e. Toset  ->  K  e.  Poset )
2 tleile.b . . . 4  |-  B  =  ( Base `  K
)
3 tleile.l . . . 4  |-  .<_  =  ( le `  K )
4 tltnle.s . . . 4  |-  .<  =  ( lt `  K )
52, 3, 4pltval3 15921 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
61, 5syl3an1 1263 . 2  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
72, 3tleile 28101 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  \/  Y  .<_  X ) )
8 ibar 502 . . . 4  |-  ( ( X  .<_  Y  \/  Y  .<_  X )  -> 
( -.  Y  .<_  X  <-> 
( ( X  .<_  Y  \/  Y  .<_  X )  /\  -.  Y  .<_  X ) ) )
9 pm5.61 711 . . . 4  |-  ( ( ( X  .<_  Y  \/  Y  .<_  X )  /\  -.  Y  .<_  X )  <-> 
( X  .<_  Y  /\  -.  Y  .<_  X ) )
108, 9syl6rbb 262 . . 3  |-  ( ( X  .<_  Y  \/  Y  .<_  X )  -> 
( ( X  .<_  Y  /\  -.  Y  .<_  X )  <->  -.  Y  .<_  X ) )
117, 10syl 17 . 2  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  -.  Y  .<_  X )  <->  -.  Y  .<_  X ) )
126, 11bitrd 253 1  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  -.  Y  .<_  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916   Posetcpo 15893   ltcplt 15894  Tosetctos 15987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-preset 15881  df-poset 15899  df-plt 15912  df-toset 15988
This theorem is referenced by:  tlt2  28104  toslublem  28107  tosglblem  28109  isarchi2  28181  archirng  28184  archiabllem2c  28191  archiabl  28194
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