Theorem ltrelxr 9669

Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltrelxr

Proof of Theorem ltrelxr

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

Step	Hyp	Ref	Expression
1		df-ltxr 9654	. 2
2		df-3an 975	. . . . . 6
3	2	opabbii 4516	. . . . 5
4		opabssxp 5079	. . . . 5
5	3, 4	eqsstri 3533	. . . 4
6		rexpssxrxp 9659	. . . 4
7	5, 6	sstri 3512	. . 3
8		ressxr 9658	. . . . . 6
9		snsspr2 4180	. . . . . . 7
10		ssun2 3667	. . . . . . . 8
11		df-xr 9653	. . . . . . . 8
12	10, 11	sseqtr4i 3536	. . . . . . 7
13	9, 12	sstri 3512	. . . . . 6
14	8, 13	unssi 3678	. . . . 5
15		snsspr1 4179	. . . . . 6
16	15, 12	sstri 3512	. . . . 5
17		xpss12 5113	. . . . 5
18	14, 16, 17	mp2an 672	. . . 4
19		xpss12 5113	. . . . 5
20	13, 8, 19	mp2an 672	. . . 4
21	18, 20	unssi 3678	. . 3
22	7, 21	unssi 3678	. 2
23	1, 22	eqsstri 3533	1

Syntax hints:  /\wa 369  /\w3a 973  e.wcel 1818  u.cun 3473  C_wss 3475  {csn 4029  {cpr 4031   class class class wbr 4452  {copab 4509  X.cxp 5002  cr 9512  cltrr 9517  cpnf 9646  cmnf 9647  cxr 9648  clt 9649

This theorem is referenced by:  ltrel  9670  dfle2  11382  dflt2  11383  itg2gt0cn  30070

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-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-v 3111  df-un 3480  df-in 3482  df-ss 3489  df-pr 4032  df-opab 4511  df-xp 5010  df-xr 9653  df-ltxr 9654