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Theorem ltxr 8695

Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
ltxr	$/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Proof of Theorem ltxr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 3769	. . . . 5 $/\$
2		df-3an 887	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 3824	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4415	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 3810	. . . 4 $\/$
7		brxp 4375	. . . . . . 7 $/\$
8		elun 3084	. . . . . . . . . . 11 $\/$
9		orcom 647	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 173	. . . . . . . . . 10 $\/$
11		elsng 3390	. . . . . . . . . . 11
12	11	orbi1d 705	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 181	. . . . . . . . 9 $\/$
14		elsng 3390	. . . . . . . . 9
15	13, 14	bi2anan9 538	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 732	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	syl6bb 185	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 181	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4375	. . . . . . 7 $/\$
20	11	anbi1d 438	. . . . . . . 8 $/\$ $/\$
21	20	adantr 261	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 181	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 707	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 684	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	syl6bb 185	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 181	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 707	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7065	. . . 4 $/\$ $/\$
29	28	breqi 3770	. . 3 $/\$ $/\$
30		brun 3810	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 173	. 2 $/\$ $/\$ $\/$
32		orass 684	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 212	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $/\$ w3a 885

wceq 1243

wcel 1393

cun 2915

csn 3375 class class class wbr 3764

copab 3817

cxp 4343

cr 6888

cltrr 6893

cpnf 7057

cmnf 7058

cxr 7059

clt 7060

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-ltxr 7065

This theorem is referenced by: xrltnr 8701 ltpnf 8702 mnflt 8704 mnfltpnf 8706 pnfnlt 8708 nltmnf 8709

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