xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9971

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 9968	. 2 $\/$ $\/$
2		ltnr 8219	. . 3
3		pnfnre 8184	. . . . . . . . . 10
4	3	neli 2497	. . . . . . . . 9
5	4	intnan 934	. . . . . . . 8 $/\$
6	5	intnanr 935	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8197	. . . . . . . . 9
8	7	neii 2402	. . . . . . . 8
9	8	intnanr 935	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 818	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 935	. . . . . . 7 $/\$
12	4	intnan 934	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 818	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 818	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8195	. . . . . 6
16		ltxr 9967	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 675	. . . 4
19		breq12 4087	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 679	. . 3
22		mnfnre 8185	. . . . . . . . . 10
23	22	neli 2497	. . . . . . . . 9
24	23	intnan 934	. . . . . . . 8 $/\$
25	24	intnanr 935	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2446	. . . . . . . 8
27	26	intnan 934	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 818	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 935	. . . . . . 7 $/\$
30	23	intnan 934	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 818	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 818	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8199	. . . . . 6
34		ltxr 9967	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 675	. . . 4
37		breq12 4087	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 679	. . 3
40	2, 21, 39	3jaoi 1337	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 713 $\/$ w3o 1001

wceq 1395

wcel 2200 class class class wbr 4082

cr 7994

cltrr 7999

cpnf 8174

cmnf 8175

cxr 8176

clt 8177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-pre-ltirr 8107

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-opab 4145 df-xp 4724 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182

This theorem is referenced by: xrltnsym 9985 xrltso 9988 xrlttri3 9989 xrleid 9992 xrltne 10005 nltpnft 10006 ngtmnft 10009 xrrebnd 10011 xposdif 10074 lbioog 10105 ubioog 10106 xrmaxleim 11750 xrmaxiflemlub 11754