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Theorem tlt3 26262
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 2451 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 tlt3.l . . . 4  |-  .<  =  ( lt `  K )
41, 2, 3tlt2 26261 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  \/  Y  .<  X ) )
5 tospos 26255 . . . . 5  |-  ( K  e. Toset  ->  K  e.  Poset )
61, 2, 3pleval2 15239 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
7 orcom 387 . . . . . 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 1252 . . . 4  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  <->  ( X  =  Y  \/  X  .<  Y ) ) )
109orbi1d 702 . . 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 966 . 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 368    \/ w3o 964    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4392   ` cfv 5518   Basecbs 14278   lecple 14349   Posetcpo 15214   ltcplt 15215  Tosetctos 15307
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-poset 15220  df-plt 15232  df-toset 15308
This theorem is referenced by:  archirngz  26342  archiabllem1b  26345  archiabllem2b  26349
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