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Theorem posasymb 14399
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 14397 . . . 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 14398 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
10 breq2 4208 . . . . 5  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
119, 10syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
12 breq1 4207 . . . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13459   lecple 13526   Posetcpo 14387
This theorem is referenced by:  pltnle  14413  pltval3  14414  lubid  14429  latasymb  14473  latleeqj1  14482  latleeqm1  14498  odupos  14552  poslubmo  14563  ople0  29886  op1le  29891  atlle0  30004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-poset 14393
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