ltxr - Intuitionistic Logic Explorer

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Theorem ltxr 9896

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 4048	. . . . 5 $/\$
2		df-3an 982	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4110	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4749	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4094	. . . 4 $\/$
7		brxp 4705	. . . . . . 7 $/\$
8		elun 3313	. . . . . . . . . . 11 $\/$
9		orcom 729	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 184	. . . . . . . . . 10 $\/$
11		elsng 3647	. . . . . . . . . . 11
12	11	orbi1d 792	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	bitrid 192	. . . . . . . . 9 $\/$
14		elsng 3647	. . . . . . . . 9
15	13, 14	bi2anan9 606	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 820	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 196	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	bitrid 192	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4705	. . . . . . 7 $/\$
20	11	anbi1d 465	. . . . . . . 8 $/\$ $/\$
21	20	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	bitrid 192	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 794	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 768	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 196	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	bitrid 192	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 794	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 8111	. . . 4 $/\$ $/\$
29	28	breqi 4049	. . 3 $/\$ $/\$
30		brun 4094	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 184	. 2 $/\$ $/\$ $\/$
32		orass 768	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 223	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $/\$ w3a 980

wceq 1372

wcel 2175

cun 3163

csn 3632 class class class wbr 4043

copab 4103

cxp 4672

cr 7923

cltrr 7928

cpnf 8103

cmnf 8104

cxr 8105

clt 8106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-opab 4105 df-xp 4680 df-ltxr 8111

This theorem is referenced by: xrltnr 9900 ltpnf 9901 mnflt 9904 mnfltpnf 9906 pnfnlt 9908 nltmnf 9909

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