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Theorem posref 14408
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 20 . . . 4  |-  ( X  e.  B  ->  X  e.  B )
21, 1, 13jca 1134 . . 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 14407 . . 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 461 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  .<_  X  /\  (
( X  .<_  X  /\  X  .<_  X )  ->  X  =  X )  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
76simp1d 969 1  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397
This theorem is referenced by:  posasymb  14409  pleval2  14422  pltval3  14424  pospo  14430  lubid  14439  latref  14482  odupos  14562  cvrnbtwn2  30073  cvrnbtwn3  30074  cvrnbtwn4  30077  cvrcmp  30081  llncmp  30319  lplncmp  30359  lvolcmp  30414
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 2417  ax-nul 4338
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 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-poset 14403
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