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Theorem ltxr 9732

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 3994	. . . . 5 $/\$
2		df-3an 975	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4056	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4686	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4040	. . . 4 $\/$
7		brxp 4642	. . . . . . 7 $/\$
8		elun 3268	. . . . . . . . . . 11 $\/$
9		orcom 723	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 183	. . . . . . . . . 10 $\/$
11		elsng 3598	. . . . . . . . . . 11
12	11	orbi1d 786	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 191	. . . . . . . . 9 $\/$
14		elsng 3598	. . . . . . . . 9
15	13, 14	bi2anan9 601	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 814	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 195	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 191	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4642	. . . . . . 7 $/\$
20	11	anbi1d 462	. . . . . . . 8 $/\$ $/\$
21	20	adantr 274	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 191	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 788	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 762	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 195	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 191	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 788	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7959	. . . 4 $/\$ $/\$
29	28	breqi 3995	. . 3 $/\$ $/\$
30		brun 4040	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 183	. 2 $/\$ $/\$ $\/$
32		orass 762	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 222	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $/\$ w3a 973

wceq 1348

wcel 2141

cun 3119

csn 3583 class class class wbr 3989

copab 4049

cxp 4609

cr 7773

cltrr 7778

cpnf 7951

cmnf 7952

cxr 7953

clt 7954

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-ltxr 7959

This theorem is referenced by: xrltnr 9736 ltpnf 9737 mnflt 9740 mnfltpnf 9742 pnfnlt 9744 nltmnf 9745

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