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Theorem ltxr 9897

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 4049	. . . . 5 $/\$
2		df-3an 983	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4111	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4750	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4095	. . . 4 $\/$
7		brxp 4706	. . . . . . 7 $/\$
8		elun 3314	. . . . . . . . . . 11 $\/$
9		orcom 730	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 184	. . . . . . . . . 10 $\/$
11		elsng 3648	. . . . . . . . . . 11
12	11	orbi1d 793	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	bitrid 192	. . . . . . . . 9 $\/$
14		elsng 3648	. . . . . . . . 9
15	13, 14	bi2anan9 606	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 821	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 196	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	bitrid 192	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4706	. . . . . . 7 $/\$
20	11	anbi1d 465	. . . . . . . 8 $/\$ $/\$
21	20	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	bitrid 192	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 795	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 769	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 196	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	bitrid 192	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 795	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 8112	. . . 4 $/\$ $/\$
29	28	breqi 4050	. . 3 $/\$ $/\$
30		brun 4095	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 184	. 2 $/\$ $/\$ $\/$
32		orass 769	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 223	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 $/\$ w3a 981

wceq 1373

wcel 2176

cun 3164

csn 3633 class class class wbr 4044

copab 4104

cxp 4673

cr 7924

cltrr 7929

cpnf 8104

cmnf 8105

cxr 8106

clt 8107

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-br 4045 df-opab 4106 df-xp 4681 df-ltxr 8112

This theorem is referenced by: xrltnr 9901 ltpnf 9902 mnflt 9905 mnfltpnf 9907 pnfnlt 9909 nltmnf 9910

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