Theorem somo 4839

Description: A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
somo

Distinct variable groups:   ,,   ,,

Proof of Theorem somo

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4455	. . . . . . . . . . 11
2	1	notbid 294	. . . . . . . . . 10
3	2	rspcv 3206	. . . . . . . . 9
4		breq1 4455	. . . . . . . . . . 11
5	4	notbid 294	. . . . . . . . . 10
6	5	rspcv 3206	. . . . . . . . 9
7	3, 6	im2anan9 835	. . . . . . . 8
8	7	ancomsd 454	. . . . . . 7
9	8	imp 429	. . . . . 6
10		ioran 490	. . . . . . 7
11		solin 4828	. . . . . . . . 9
12		df-3or 974	. . . . . . . . . 10
13		or32 527	. . . . . . . . . 10
14	12, 13	bitri 249	. . . . . . . . 9
15	11, 14	sylib 196	. . . . . . . 8
16	15	ord 377	. . . . . . 7
17	10, 16	syl5bir 218	. . . . . 6
18	9, 17	syl5 32	. . . . 5
19	18	exp4b 607	. . . 4
20	19	pm2.43d 48	. . 3
21	20	ralrimivv 2877	. 2
22		breq2 4456	. . . . 5
23	22	notbid 294	. . . 4
24	23	ralbidv 2896	. . 3
25	24	rmo4 3292	. 2
26	21, 25	sylibr 212	1

Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  \/w3o 972  e.wcel 1818  A.wral 2807  E*wrmo 2810   class class class wbr 4452  Orwor 4804

This theorem is referenced by:  wereu  4880  wereu2  4881

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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rmo 2815  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