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Theorem ltxr 11949

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 4658	. . . . 5 $/\$
2		df-3an 1039	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4717	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2a 5194	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 11	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4703	. . . 4 $\/$
7		brxp 5147	. . . . . . 7 $/\$
8		elun 3753	. . . . . . . . . . 11 $\/$
9		orcom 402	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 264	. . . . . . . . . 10 $\/$
11		elsng 4191	. . . . . . . . . . 11
12	11	orbi1d 739	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 272	. . . . . . . . 9 $\/$
14		elsng 4191	. . . . . . . . 9
15	13, 14	bi2anan9 917	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 912	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	syl6bb 276	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 272	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 5147	. . . . . . 7 $/\$
20	11	anbi1d 741	. . . . . . . 8 $/\$ $/\$
21	20	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 272	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 746	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 546	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	syl6bb 276	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 272	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 746	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 10079	. . . 4 $/\$ $/\$
29	28	breqi 4659	. . 3 $/\$ $/\$
30		brun 4703	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 264	. 2 $/\$ $/\$ $\/$
32		orass 546	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 303	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cun 3572

csn 4177 class class class wbr 4653

copab 4712

cxp 5112

cr 9935

cltrr 9940

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-ltxr 10079

This theorem is referenced by: xrltnr 11953 ltpnf 11954 mnflt 11957 mnfltpnf 11960 pnfnlt 11962 nltmnf 11963

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