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Theorem prsref 16255
Description: Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b  |-  B  =  ( Base `  K
)
isprs.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
prsref  |-  ( ( K  e.  Preset  /\  X  e.  B )  ->  X  .<_  X )

Proof of Theorem prsref
StepHypRef Expression
1 id 22 . . . 4  |-  ( X  e.  B  ->  X  e.  B )
21, 1, 13jca 1210 . . 3  |-  ( X  e.  B  ->  ( X  e.  B  /\  X  e.  B  /\  X  e.  B )
)
3 isprs.b . . . 4  |-  B  =  ( Base `  K
)
4 isprs.l . . . 4  |-  .<_  =  ( le `  K )
53, 4prslem 16254 . . 3  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  X  e.  B  /\  X  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
62, 5sylan2 482 . 2  |-  ( ( K  e.  Preset  /\  X  e.  B )  ->  ( X  .<_  X  /\  (
( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
76simpld 466 1  |-  ( ( K  e.  Preset  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275    Preset cpreset 16249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-preset 16251
This theorem is referenced by:  posref  16274  prsdm  28794  prsrn  28795
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