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Theorem posref 14101
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 14100 . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090
This theorem is referenced by:  posasymb  14102  pleval2  14115  pltval3  14117  pospo  14123  lubid  14132  latref  14175  odupos  14255  cvrnbtwn2  30087  cvrnbtwn3  30088  cvrnbtwn4  30091  cvrcmp  30095  llncmp  30333  lplncmp  30373  lvolcmp  30428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-poset 14096
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