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Theorem posasymb 15430
Description: A poset ordering is asymmetric. (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 991 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
2 simp2 992 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 simp3 993 . . . 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 15428 . . . 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 1225 . . 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 1004 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
94, 5posref 15429 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
10 breq2 4446 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
119, 10syl5ibcom 220 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
12 breq1 4445 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
139, 12syl5ibcom 220 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
1411, 13jcad 533 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
15143adant3 1011 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
168, 15impbid 191 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   class class class wbr 4442   ` 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-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
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-iota 5544  df-fv 5589  df-poset 15424
This theorem is referenced by:  pltnle  15444  pltval3  15445  lublecllem  15466  latasymb  15532  latleeqj1  15541  latleeqm1  15557  odupos  15613  poslubmo  15624  posglbmo  15625  posrasymb  27295  archirngz  27383  archiabllem1a  27385  ople0  33861  op1le  33866  atlle0  33979
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