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Theorem trleile 28106
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
Hypotheses
Ref Expression
trleile.b  |-  B  =  ( Base `  K
)
trleile.l  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
Assertion
Ref Expression
trleile  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  \/  Y  .<_  X ) )

Proof of Theorem trleile
StepHypRef Expression
1 trleile.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2402 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
31, 2tleile 28101 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le `  K ) Y  \/  Y ( le `  K ) X ) )
4 3simpc 996 . . . . . 6  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  e.  B  /\  Y  e.  B )
)
5 brxp 4854 . . . . . 6  |-  ( X ( B  X.  B
) Y  <->  ( X  e.  B  /\  Y  e.  B ) )
64, 5sylibr 212 . . . . 5  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  X
( B  X.  B
) Y )
7 brin 4444 . . . . . 6  |-  ( X ( ( le `  K )  i^i  ( B  X.  B ) ) Y  <->  ( X ( le `  K ) Y  /\  X ( B  X.  B ) Y ) )
87rbaib 907 . . . . 5  |-  ( X ( B  X.  B
) Y  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  <->  X ( le `  K ) Y ) )
96, 8syl 17 . . . 4  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  <->  X ( le `  K ) Y ) )
104ancomd 449 . . . . . 6  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  e.  B  /\  X  e.  B )
)
11 brxp 4854 . . . . . 6  |-  ( Y ( B  X.  B
) X  <->  ( Y  e.  B  /\  X  e.  B ) )
1210, 11sylibr 212 . . . . 5  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  Y
( B  X.  B
) X )
13 brin 4444 . . . . . 6  |-  ( Y ( ( le `  K )  i^i  ( B  X.  B ) ) X  <->  ( Y ( le `  K ) X  /\  Y ( B  X.  B ) X ) )
1413rbaib 907 . . . . 5  |-  ( Y ( B  X.  B
) X  ->  ( Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X  <->  Y ( le `  K ) X ) )
1512, 14syl 17 . . . 4  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X  <->  Y ( le `  K ) X ) )
169, 15orbi12d 708 . . 3  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X ( ( le `  K )  i^i  ( B  X.  B ) ) Y  \/  Y ( ( le `  K )  i^i  ( B  X.  B ) ) X )  <->  ( X ( le `  K ) Y  \/  Y ( le `  K ) X ) ) )
173, 16mpbird 232 . 2  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  \/  Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X ) )
18 trleile.l . . . 4  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
1918breqi 4401 . . 3  |-  ( X 
.<_  Y  <->  X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y )
2018breqi 4401 . . 3  |-  ( Y 
.<_  X  <->  Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X )
2119, 20orbi12i 519 . 2  |-  ( ( X  .<_  Y  \/  Y  .<_  X )  <->  ( X
( ( le `  K )  i^i  ( B  X.  B ) ) Y  \/  Y ( ( le `  K
)  i^i  ( B  X.  B ) ) X ) )
2217, 21sylibr 212 1  |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  \/  Y  .<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3413   class class class wbr 4395    X. cxp 4821   ` cfv 5569   Basecbs 14841   lecple 14916  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-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-xp 4829  df-iota 5533  df-fv 5577  df-toset 15988
This theorem is referenced by:  ordtrest2NEWlem  28357  ordtconlem1  28359
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