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Theorem posasymb 14102
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 955 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
2 simp2 956 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 simp3 957 . . . 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 14100 . . . 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 1184 . . 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 968 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
94, 5posref 14101 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
10 breq2 4043 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
119, 10syl5ibcom 211 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
12 breq1 4042 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
139, 12syl5ibcom 211 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
1411, 13jcad 519 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
15143adant3 975 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
168, 15impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090
This theorem is referenced by:  pltnle  14116  pltval3  14117  lubid  14132  latasymb  14176  latleeqj1  14185  latleeqm1  14201  odupos  14255  poslubmo  14266  ople0  29999  op1le  30004  atlle0  30117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-poset 14096
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