ltxr - Intuitionistic Logic Explorer

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Theorem ltxr 9702

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 3981	. . . . 5 $/\$
2		df-3an 969	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4043	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4673	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4027	. . . 4 $\/$
7		brxp 4629	. . . . . . 7 $/\$
8		elun 3258	. . . . . . . . . . 11 $\/$
9		orcom 718	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 183	. . . . . . . . . 10 $\/$
11		elsng 3585	. . . . . . . . . . 11
12	11	orbi1d 781	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	syl5bb 191	. . . . . . . . 9 $\/$
14		elsng 3585	. . . . . . . . 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 4629	. . . . . . 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 7929	. . . 4 $/\$ $/\$
29	28	breqi 3982	. . 3 $/\$ $/\$
30		brun 4027	. . 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 967

wceq 1342

wcel 2135

cun 3109

csn 3570 class class class wbr 3976

copab 4036

cxp 4596

cr 7743

cltrr 7748

cpnf 7921

cmnf 7922

cxr 7923

clt 7924

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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-ltxr 7929

This theorem is referenced by: xrltnr 9706 ltpnf 9707 mnflt 9710 mnfltpnf 9712 pnfnlt 9714 nltmnf 9715

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