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Theorem posasymb 14329
Description: A poset ordering is asymetric. (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
posi.b  |-  B  =  ( Base `  K
)
posi.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
posasymb  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )

Proof of Theorem posasymb
StepHypRef Expression
1 simp1 957 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
2 simp2 958 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 simp3 959 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
4 posi.b . . . . 5  |-  B  =  ( Base `  K
)
5 posi.l . . . . 5  |-  .<_  =  ( le `  K )
64, 5posi 14327 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )  /\  ( ( X  .<_  Y  /\  Y  .<_  Y )  ->  X  .<_  Y ) ) )
71, 2, 3, 3, 6syl13anc 1186 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  X  /\  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )  /\  ( ( X  .<_  Y  /\  Y  .<_  Y )  ->  X  .<_  Y ) ) )
87simp2d 970 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
94, 5posref 14328 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
10 breq2 4150 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
119, 10syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
12 breq1 4149 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
139, 12syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
1411, 13jcad 520 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
15143adant3 977 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
168, 15impbid 184 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   Posetcpo 14317
This theorem is referenced by:  pltnle  14343  pltval3  14344  lubid  14359  latasymb  14403  latleeqj1  14412  latleeqm1  14428  odupos  14482  poslubmo  14493  ople0  29353  op1le  29358  atlle0  29471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-poset 14323
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