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Theorem ltxr 9762

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 4005	. . . . 5 $/\$
2		df-3an 980	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4067	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4698	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4051	. . . 4 $\/$
7		brxp 4654	. . . . . . 7 $/\$
8		elun 3276	. . . . . . . . . . 11 $\/$
9		orcom 728	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 184	. . . . . . . . . 10 $\/$
11		elsng 3606	. . . . . . . . . . 11
12	11	orbi1d 791	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	bitrid 192	. . . . . . . . 9 $\/$
14		elsng 3606	. . . . . . . . 9
15	13, 14	bi2anan9 606	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 819	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 196	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	bitrid 192	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4654	. . . . . . 7 $/\$
20	11	anbi1d 465	. . . . . . . 8 $/\$ $/\$
21	20	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	bitrid 192	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 793	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 767	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 196	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	bitrid 192	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 793	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7987	. . . 4 $/\$ $/\$
29	28	breqi 4006	. . 3 $/\$ $/\$
30		brun 4051	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 184	. 2 $/\$ $/\$ $\/$
32		orass 767	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 223	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2148

cun 3127

csn 3591 class class class wbr 4000

copab 4060

cxp 4621

cr 7801

cltrr 7806

cpnf 7979

cmnf 7980

cxr 7981

clt 7982

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-ltxr 7987

This theorem is referenced by: xrltnr 9766 ltpnf 9767 mnflt 9770 mnfltpnf 9772 pnfnlt 9774 nltmnf 9775

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