Theorem sotri 5399

Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1
soi.2

Assertion

Ref	Expression
sotri

Proof of Theorem sotri

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5
2	1	brel 5053	. . . 4
3	2	simpld 459	. . 3
4	1	brel 5053	. . 3
5	3, 4	anim12i 566	. 2
6		soi.1	. . . 4
7		sotr 4827	. . . 4
8	6, 7	mpan 670	. . 3
9	8	3expb 1197	. 2
10	5, 9	mpcom 36	1

Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  e.wcel 1818  C_wss 3475   class class class wbr 4452  Orwor 4804  X.cxp 5002

This theorem is referenced by:  son2lpi  5400  sotri2  5401  sotri3  5402  son2lpiOLD  5405  ltsonq  9368  ltbtwnnq  9377  nqpr  9413  prlem934  9432  ltexprlem4  9438  reclem2pr  9447  reclem4pr  9449  ltsosr  9492  addgt0sr  9502  supsrlem  9509  axpre-lttrn  9564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-po 4805  df-so 4806  df-xp 5010