MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pltn2lp Structured version   Unicode version

Theorem pltn2lp 14418
Description: The less-than relation has no 2-cycle loops. (pssn2lp 3440 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 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 pltnlt.s . . . . 5  |-  .<  =  ( lt `  K )
41, 2, 3pltnle 14415 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y ( le `  K ) X )
54ex 424 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y ( le `  K ) X ) )
62, 3pltle 14410 . . . 4  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
763com23 1159 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
85, 7nsyld 134 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<  X ) )
9 imnan 412 . 2  |-  ( ( X  .<  Y  ->  -.  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  X ) )
108, 9sylib 189 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   ltcplt 14390
This theorem is referenced by:  plttr  14419
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-mo 2285  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-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-poset 14395  df-plt 14407
  Copyright terms: Public domain W3C validator