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Theorem posrasymb 27295
Description: A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
posrasymb.b  |-  B  =  ( Base `  K
)
posrasymb.l  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
Assertion
Ref Expression
posrasymb  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )

Proof of Theorem posrasymb
StepHypRef Expression
1 posrasymb.l . . . . 5  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
21breqi 4448 . . . 4  |-  ( X 
.<_  Y  <->  X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y )
3 simp2 992 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 simp3 993 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
5 brxp 5024 . . . . . 6  |-  ( X ( B  X.  B
) Y  <->  ( X  e.  B  /\  Y  e.  B ) )
63, 4, 5sylanbrc 664 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X
( B  X.  B
) Y )
7 brin 4491 . . . . . 6  |-  ( X ( ( le `  K )  i^i  ( B  X.  B ) ) Y  <->  ( X ( le `  K ) Y  /\  X ( B  X.  B ) Y ) )
87rbaib 900 . . . . 5  |-  ( X ( B  X.  B
) Y  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  <->  X ( le `  K ) Y ) )
96, 8syl 16 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  <->  X ( le `  K ) Y ) )
102, 9syl5bb 257 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X ( le `  K ) Y ) )
111breqi 4448 . . . 4  |-  ( Y 
.<_  X  <->  Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X )
12 brxp 5024 . . . . . 6  |-  ( Y ( B  X.  B
) X  <->  ( Y  e.  B  /\  X  e.  B ) )
134, 3, 12sylanbrc 664 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  Y
( B  X.  B
) X )
14 brin 4491 . . . . . 6  |-  ( Y ( ( le `  K )  i^i  ( B  X.  B ) ) X  <->  ( Y ( le `  K ) X  /\  Y ( B  X.  B ) X ) )
1514rbaib 900 . . . . 5  |-  ( Y ( B  X.  B
) X  ->  ( Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X  <->  Y ( le `  K ) X ) )
1613, 15syl 16 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X  <->  Y ( le `  K ) X ) )
1711, 16syl5bb 257 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  Y ( le `  K ) X ) )
1810, 17anbi12d 710 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  ( X
( le `  K
) Y  /\  Y
( le `  K
) X ) ) )
19 posrasymb.b . . 3  |-  B  =  ( Base `  K
)
20 eqid 2462 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2119, 20posasymb 15430 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X ( le
`  K ) Y  /\  Y ( le
`  K ) X )  <->  X  =  Y
) )
2218, 21bitrd 253 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    i^i cin 3470   class class class wbr 4442    X. cxp 4992   ` cfv 5581   Basecbs 14481   lecple 14553   Posetcpo 15418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-iota 5544  df-fv 5589  df-poset 15424
This theorem is referenced by:  ordtconlem1  27530
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