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Theorem prsref 14381
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 14380 . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528    Preset cpreset 14375
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 2416  ax-nul 4330
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-preset 14377
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