xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 8701

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 8696	. 2 $\/$ $\/$
2		ltnr 7095	. . 3
3		pnfnre 7067	. . . . . . . . . 10
4	3	neli 2299	. . . . . . . . 9
5	4	intnan 838	. . . . . . . 8 $/\$
6	5	intnanr 839	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8697	. . . . . . . . 9
8	7	neii 2208	. . . . . . . 8
9	8	intnanr 839	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 726	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 839	. . . . . . 7 $/\$
12	4	intnan 838	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 726	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 726	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8692	. . . . . 6
16		ltxr 8695	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 402	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 596	. . . 4
19		breq12 3769	. . . . 5 $/\$
20	19	anidms 377	. . . 4
21	18, 20	mtbiri 600	. . 3
22		mnfnre 7068	. . . . . . . . . 10
23	22	neli 2299	. . . . . . . . 9
24	23	intnan 838	. . . . . . . 8 $/\$
25	24	intnanr 839	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2251	. . . . . . . 8
27	26	intnan 838	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 726	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 839	. . . . . . 7 $/\$
30	23	intnan 838	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 726	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 726	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8694	. . . . . 6
34		ltxr 8695	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 402	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 596	. . . 4
37		breq12 3769	. . . . 5 $/\$
38	37	anidms 377	. . . 4
39	36, 38	mtbiri 600	. . 3
40	2, 21, 39	3jaoi 1198	. 2 $\/$ $\/$
41	1, 40	sylbi 114	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $\/$ w3o 884

wceq 1243

wcel 1393 class class class wbr 3764

cr 6888

cltrr 6893

cpnf 7057

cmnf 7058

cxr 7059

clt 7060

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996

This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065

This theorem is referenced by: xrltnsym 8714 xrltso 8717 xrlttri3 8718 xrleid 8720 xrltne 8729 nltpnft 8730 ngtmnft 8731 xrrebnd 8732 lbioog 8782 ubioog 8783