ltxr - Intuitionistic Logic Explorer

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Theorem ltxr 8849

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 3790	. . . . 5 $/\$
2		df-3an 921	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 3845	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4433	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 3831	. . . 4 $\/$
7		brxp 4393	. . . . . . 7 $/\$
8		elun 3113	. . . . . . . . . . 11 $\/$
9		orcom 679	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 182	. . . . . . . . . 10 $\/$
11		elsng 3413	. . . . . . . . . . 11
12	11	orbi1d 737	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 190	. . . . . . . . 9 $\/$
14		elsng 3413	. . . . . . . . 9
15	13, 14	bi2anan9 570	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 765	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	syl6bb 194	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 190	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4393	. . . . . . 7 $/\$
20	11	anbi1d 452	. . . . . . . 8 $/\$ $/\$
21	20	adantr 270	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 190	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 739	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 716	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	syl6bb 194	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 190	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 739	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7158	. . . 4 $/\$ $/\$
29	28	breqi 3791	. . 3 $/\$ $/\$
30		brun 3831	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 182	. 2 $/\$ $/\$ $\/$
32		orass 716	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 221	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661 $/\$ w3a 919

wceq 1284

wcel 1433

cun 2971

csn 3398 class class class wbr 3785

copab 3838

cxp 4361

cr 6980

cltrr 6985

cpnf 7150

cmnf 7151

cxr 7152

clt 7153

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-ltxr 7158

This theorem is referenced by: xrltnr 8855 ltpnf 8856 mnflt 8858 mnfltpnf 8860 pnfnlt 8862 nltmnf 8863

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