xrltnr - Metamath Proof Explorer

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Theorem xrltnr 11953

Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
xrltnr

**Proof of Theorem xrltnr**
Step	Hyp	Ref	Expression
1		elxr 11950	. 2 $\/$ $\/$
2		ltnr 10132	. . 3
3		pnfnre 10081	. . . . . . . . . 10
4	3	neli 2899	. . . . . . . . 9
5	4	intnan 960	. . . . . . . 8 $/\$
6	5	intnanr 961	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 10094	. . . . . . . . 9
8	7	neii 2796	. . . . . . . 8
9	8	intnanr 961	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 899	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 961	. . . . . . 7 $/\$
12	4	intnan 960	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 899	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 899	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 10092	. . . . . 6
16		ltxr 11949	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 708	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 313	. . . 4
19		breq12 4658	. . . . 5 $/\$
20	19	anidms 677	. . . 4
21	18, 20	mtbiri 317	. . 3
22		mnfnre 10082	. . . . . . . . . 10
23	22	neli 2899	. . . . . . . . 9
24	23	intnan 960	. . . . . . . 8 $/\$
25	24	intnanr 961	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2851	. . . . . . . 8
27	26	intnan 960	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 899	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 961	. . . . . . 7 $/\$
30	23	intnan 960	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 899	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 899	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 10096	. . . . . 6
34		ltxr 11949	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 708	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 313	. . . 4
37		breq12 4658	. . . . 5 $/\$
38	37	anidms 677	. . . 4
39	36, 38	mtbiri 317	. . 3
40	2, 21, 39	3jaoi 1391	. 2 $\/$ $\/$
41	1, 40	sylbi 207	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990 class class class wbr 4653

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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079

This theorem is referenced by: xrltnsym 11970 xrlttri 11972 nltpnft 11995 ngtmnft 11997 xrsupsslem 12137 xrinfmsslem 12138 xrub 12142 lbioo 12206 ubioo 12207 topnfbey 27325 lbioc 39739 icoub 39752 iccpartnel 41374