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Theorem tltnle 26295
Description: In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 15259. (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 26291 . . 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 15260 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
61, 5syl3an1 1252 . 2  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
72, 3tleile 26294 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  \/  Y  .<_  X ) )
8 ibar 504 . . . 4  |-  ( ( X  .<_  Y  \/  Y  .<_  X )  -> 
( -.  Y  .<_  X  <-> 
( ( X  .<_  Y  \/  Y  .<_  X )  /\  -.  Y  .<_  X ) ) )
9 pm5.61 712 . . . 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 16 . 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 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529   Basecbs 14296   lecple 14368   Posetcpo 15233   ltcplt 15234  Tosetctos 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-poset 15239  df-plt 15251  df-toset 15327
This theorem is referenced by:  tlt2  26297  toslublem  26300  tosglblem  26302  isarchi2  26374  archirng  26377  archiabllem2c  26384  archiabl  26387
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