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Theorem tlt3 27890
Description: In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
tlt3.b  |-  B  =  ( Base `  K
)
tlt3.l  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
tlt3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  \/  X  .<  Y  \/  Y  .<  X ) )

Proof of Theorem tlt3
StepHypRef Expression
1 tlt3.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 tlt3.l . . . 4  |-  .<  =  ( lt `  K )
41, 2, 3tlt2 27889 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  \/  Y  .<  X ) )
5 tospos 27883 . . . . 5  |-  ( K  e. Toset  ->  K  e.  Poset )
61, 2, 3pleval2 15797 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
7 orcom 385 . . . . . 6  |-  ( ( X  .<  Y  \/  X  =  Y )  <->  ( X  =  Y  \/  X  .<  Y ) )
86, 7syl6bb 261 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  <->  ( X  =  Y  \/  X  .<  Y ) ) )
95, 8syl3an1 1259 . . . 4  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  <->  ( X  =  Y  \/  X  .<  Y ) ) )
109orbi1d 700 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X ( le
`  K ) Y  \/  Y  .<  X )  <-> 
( ( X  =  Y  \/  X  .<  Y )  \/  Y  .<  X ) ) )
114, 10mpbid 210 . 2  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  =  Y  \/  X  .<  Y )  \/  Y  .<  X ) )
12 df-3or 972 . 2  |-  ( ( X  =  Y  \/  X  .<  Y  \/  Y  .<  X )  <->  ( ( X  =  Y  \/  X  .<  Y )  \/  Y  .<  X )
)
1311, 12sylibr 212 1  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  \/  X  .<  Y  \/  Y  .<  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570   Basecbs 14719   lecple 14794   Posetcpo 15771   ltcplt 15772  Tosetctos 15865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-preset 15759  df-poset 15777  df-plt 15790  df-toset 15866
This theorem is referenced by:  archirngz  27970  archiabllem1b  27973  archiabllem2b  27977
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