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Theorem pltn2lp 15455
Description: The less-than relation has no 2-cycle loops. (pssn2lp 3605 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltnlt.b  |-  B  =  ( Base `  K
)
pltnlt.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltn2lp  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )

Proof of Theorem pltn2lp
StepHypRef Expression
1 pltnlt.b . . . . 5  |-  B  =  ( Base `  K
)
2 eqid 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 pltnlt.s . . . . 5  |-  .<  =  ( lt `  K )
41, 2, 3pltnle 15452 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y ( le `  K ) X )
54ex 434 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y ( le `  K ) X ) )
62, 3pltle 15447 . . . 4  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
763com23 1202 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
85, 7nsyld 140 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<  X ) )
9 imnan 422 . 2  |-  ( ( X  .<  Y  ->  -.  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  X ) )
108, 9sylib 196 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5587   Basecbs 14489   lecple 14561   Posetcpo 15426   ltcplt 15427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-poset 15432  df-plt 15444
This theorem is referenced by:  plttr  15456
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