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Theorem posasymb 16276
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 1030 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
2 simp2 1031 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 simp3 1032 . . . 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 16273 . . . 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 1294 . . 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 1043 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
94, 5posref 16274 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
10 breq2 4399 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
119, 10syl5ibcom 228 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
12 breq1 4398 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
139, 12syl5ibcom 228 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
1411, 13jcad 542 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
15143adant3 1050 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
168, 15impbid 195 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-preset 16251  df-poset 16269
This theorem is referenced by:  pltnle  16290  pltval3  16291  lublecllem  16312  latasymb  16378  latleeqj1  16387  latleeqm1  16403  odupos  16459  poslubmo  16470  posglbmo  16471  posrasymb  28493  archirngz  28580  archiabllem1a  28582  ople0  32824  op1le  32829  atlle0  32942
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