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Theorem postr 15907
Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
posi.b  |-  B  =  ( Base `  K
)
posi.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
postr  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )

Proof of Theorem postr
StepHypRef Expression
1 posi.b . . 3  |-  B  =  ( Base `  K
)
2 posi.l . . 3  |-  .<_  =  ( le `  K )
31, 2posi 15903 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
43simp3d 1011 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916   Posetcpo 15893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-poset 15899
This theorem is referenced by:  plttr  15924  joinle  15968  meetle  15982  lattr  16010  odupos  16089  omndadd2d  28150  omndadd2rd  28151  omndmul2  28154  atlatle  32338  cvratlem  32438  llncmp  32539  llncvrlpln  32575  lplncmp  32579  lplncvrlvol  32633  lvolcmp  32634  pmaple  32778
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