Theorem prstr 16933

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

Step	Hyp	Ref	Expression
1		isprs.b	. . . 4 |- B = ( Base ` K )
2		isprs.l	. . . 4 |- .<_ = ( le ` K )
3	1, 2	prslem 16931	. . 3 |- ( ( K e. Preset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ X /\ ( ( X .<_ Y /\ Y .<_ Z ) -> X .<_ Z ) ) )
4	3	simprd 479	. 2 |- ( ( K e. Preset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ Y .<_ Z ) -> X .<_ Z ) )
5	4	3impia 1261	1 |- ( ( K e. Preset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X .<_ Y /\ Y .<_ Z ) ) -> X .<_ Z )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-preset 16928

This theorem is referenced by:  drsdirfi  16938