Theorem solin 4828

Description: A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)

Assertion

Ref	Expression
solin

Proof of Theorem solin

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4455	. . . . 5
2		eqeq1 2461	. . . . 5
3		breq2 4456	. . . . 5
4	1, 2, 3	3orbi123d 1298	. . . 4
5	4	imbi2d 316	. . 3
6		breq2 4456	. . . . 5
7		eqeq2 2472	. . . . 5
8		breq1 4455	. . . . 5
9	6, 7, 8	3orbi123d 1298	. . . 4
10	9	imbi2d 316	. . 3
11		df-so 4806	. . . . 5
12		rsp2 2831	. . . . . 6
13	12	adantl 466	. . . . 5
14	11, 13	sylbi 195	. . . 4
15	14	com12 31	. . 3
16	5, 10, 15	vtocl2ga 3175	. 2
17	16	impcom 430	1

Syntax hints:  ->wi 4  /\wa 369  \/w3o 972  =wceq 1395  e.wcel 1818  A.wral 2807   class class class wbr 4452  Powpo 4803  Orwor 4804

This theorem is referenced by:  sotric  4831  sotrieq  4832  somo  4839  wecmpep  4876  sorpssi  6586  soxp  6913  wemaplem2  7993  fpwwe2lem12  9040  fpwwe2lem13  9041  lttri4  9690  xmullem  11485  xmulasslem  11506  orngsqr  27794  socnv  29194  wfrlem10  29352  slttri  29433  fin2so  30040  fnwe2lem3  30998

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-ral 2812  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-so 4806