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Theorem prsref 14309
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 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 isprs.b . . . 4  |-  B  =  ( Base `  K
)
4 isprs.l . . . 4  |-  .<_  =  ( le `  K )
53, 4prslem 14308 . . 3  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  X  e.  B  /\  X  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
62, 5sylan2 461 . 2  |-  ( ( K  e.  Preset  /\  X  e.  B )  ->  ( X  .<_  X  /\  (
( X  .<_  X  /\  X  .<_  X )  ->  X  .<_  X ) ) )
76simpld 446 1  |-  ( ( K  e.  Preset  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456    Preset cpreset 14303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-preset 14305
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