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Theorem posrasymb 28110
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 4403 . . . 4  |-  ( X 
.<_  Y  <->  X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y )
3 simp2 1000 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 simp3 1001 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
5 brxp 4856 . . . . . 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 4446 . . . . . 6  |-  ( X ( ( le `  K )  i^i  ( B  X.  B ) ) Y  <->  ( X ( le `  K ) Y  /\  X ( B  X.  B ) Y ) )
87rbaib 909 . . . . 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.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( ( le
`  K )  i^i  ( B  X.  B
) ) Y  <->  X ( le `  K ) Y ) )
102, 9syl5bb 259 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X ( le `  K ) Y ) )
111breqi 4403 . . . 4  |-  ( Y 
.<_  X  <->  Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X )
12 brxp 4856 . . . . . 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 4446 . . . . . 6  |-  ( Y ( ( le `  K )  i^i  ( B  X.  B ) ) X  <->  ( Y ( le `  K ) X  /\  Y ( B  X.  B ) X ) )
1514rbaib 909 . . . . 5  |-  ( Y ( B  X.  B
) X  ->  ( Y ( ( le
`  K )  i^i  ( B  X.  B
) ) X  <->  Y ( le `  K ) X ) )
1613, 15syl 17 . . . 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 259 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  Y ( le `  K ) X ) )
1810, 17anbi12d 711 . 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 2404 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2119, 20posasymb 15908 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X ( le
`  K ) Y  /\  Y ( le
`  K ) X )  <->  X  =  Y
) )
2218, 21bitrd 255 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    i^i cin 3415   class class class wbr 4397    X. cxp 4823   ` cfv 5571   Basecbs 14843   lecple 14918   Posetcpo 15895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-xp 4831  df-iota 5535  df-fv 5579  df-preset 15883  df-poset 15901
This theorem is referenced by:  ordtconlem1  28372
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