ltxr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ltxr		Unicode version

Theorem ltxr 9555

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 3929	. . . . 5 $/\$
2		df-3an 964	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 3990	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4609	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 3974	. . . 4 $\/$
7		brxp 4565	. . . . . . 7 $/\$
8		elun 3212	. . . . . . . . . . 11 $\/$
9		orcom 717	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 183	. . . . . . . . . 10 $\/$
11		elsng 3537	. . . . . . . . . . 11
12	11	orbi1d 780	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 191	. . . . . . . . 9 $\/$
14		elsng 3537	. . . . . . . . 9
15	13, 14	bi2anan9 595	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 808	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	syl6bb 195	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 191	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4565	. . . . . . 7 $/\$
20	11	anbi1d 460	. . . . . . . 8 $/\$ $/\$
21	20	adantr 274	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 191	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 782	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 756	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	syl6bb 195	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 191	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 782	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7798	. . . 4 $/\$ $/\$
29	28	breqi 3930	. . 3 $/\$ $/\$
30		brun 3974	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 183	. 2 $/\$ $/\$ $\/$
32		orass 756	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 222	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697 $/\$ w3a 962

wceq 1331

wcel 1480

cun 3064

csn 3522 class class class wbr 3924

copab 3983

cxp 4532

cr 7612

cltrr 7617

cpnf 7790

cmnf 7791

cxr 7792

clt 7793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-ltxr 7798

This theorem is referenced by: xrltnr 9559 ltpnf 9560 mnflt 9562 mnfltpnf 9564 pnfnlt 9566 nltmnf 9567

W3C validator