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Theorem prstr 15764
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 15762 . . 3  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
43simprd 461 . 2  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
543impia 1191 1  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Y  /\  Y  .<_  Z ) )  ->  X  .<_  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570   Basecbs 14719   lecple 14794    Preset cpreset 15757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-preset 15759
This theorem is referenced by:  drsdirfi  15769
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