ltxr - Intuitionistic Logic Explorer

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Theorem ltxr 9711

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 3987	. . . . 5 $/\$
2		df-3an 970	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4049	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4679	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4033	. . . 4 $\/$
7		brxp 4635	. . . . . . 7 $/\$
8		elun 3263	. . . . . . . . . . 11 $\/$
9		orcom 718	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 183	. . . . . . . . . 10 $\/$
11		elsng 3591	. . . . . . . . . . 11
12	11	orbi1d 781	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 191	. . . . . . . . 9 $\/$
14		elsng 3591	. . . . . . . . 9
15	13, 14	bi2anan9 596	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 809	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 195	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	syl5bb 191	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4635	. . . . . . 7 $/\$
20	11	anbi1d 461	. . . . . . . 8 $/\$ $/\$
21	20	adantr 274	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	syl5bb 191	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 783	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 757	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 195	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	syl5bb 191	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 783	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 7938	. . . 4 $/\$ $/\$
29	28	breqi 3988	. . 3 $/\$ $/\$
30		brun 4033	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 183	. 2 $/\$ $/\$ $\/$
32		orass 757	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 222	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $/\$ w3a 968

wceq 1343

wcel 2136

cun 3114

csn 3576 class class class wbr 3982

copab 4042

cxp 4602

cr 7752

cltrr 7757

cpnf 7930

cmnf 7931

cxr 7932

clt 7933

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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-br 3983 df-opab 4044 df-xp 4610 df-ltxr 7938

This theorem is referenced by: xrltnr 9715 ltpnf 9716 mnflt 9719 mnfltpnf 9721 pnfnlt 9723 nltmnf 9724

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