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Theorem prstr 14390
Description: Less-or-equal is transitive 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
prstr  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Y  /\  Y  .<_  Z ) )  ->  X  .<_  Z )

Proof of Theorem prstr
StepHypRef Expression
1 isprs.b . . . 4  |-  B  =  ( Base `  K
)
2 isprs.l . . . 4  |-  .<_  =  ( le `  K )
31, 2prslem 14388 . . 3  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
43simprd 450 . 2  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
543impia 1150 1  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Y  /\  Y  .<_  Z ) )  ->  X  .<_  Z )
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    Preset cpreset 14383
This theorem is referenced by:  drsdirfi  14395
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-preset 14385
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