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Theorem posref 14085
Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
posi.b  |-  B  =  ( Base `  K
)
posi.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
posref  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )

Proof of Theorem posref
StepHypRef Expression
1 id 19 . . . 4  |-  ( X  e.  B  ->  X  e.  B )
21, 1, 13jca 1132 . . 3  |-  ( X  e.  B  ->  ( X  e.  B  /\  X  e.  B  /\  X  e.  B )
)
3 posi.b . . . 4  |-  B  =  ( Base `  K
)
4 posi.l . . . 4  |-  .<_  =  ( le `  K )
53, 4posi 14084 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  X  e.  B  /\  X  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  =  X )  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
62, 5sylan2 460 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  .<_  X  /\  (
( X  .<_  X  /\  X  .<_  X )  ->  X  =  X )  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
76simp1d 967 1  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074
This theorem is referenced by:  posasymb  14086  pleval2  14099  pltval3  14101  pospo  14107  lubid  14116  latref  14159  odupos  14239  cvrnbtwn2  29465  cvrnbtwn3  29466  cvrnbtwn4  29469  cvrcmp  29473  llncmp  29711  lplncmp  29751  lvolcmp  29806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-poset 14080
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