xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10160

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 10157	. 2 $\/$ $\/$
2		ltnr 8392	. . 3
3		pnfnre 8357	. . . . . . . . . 10
4	3	neli 2517	. . . . . . . . 9
5	4	intnan 941	. . . . . . . 8 $/\$
6	5	intnanr 942	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8370	. . . . . . . . 9
8	7	neii 2422	. . . . . . . 8
9	8	intnanr 942	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 825	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 942	. . . . . . 7 $/\$
12	4	intnan 941	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 825	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 825	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8368	. . . . . 6
16		ltxr 10156	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 430	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 682	. . . 4
19		breq12 4130	. . . . 5 $/\$
20	19	anidms 401	. . . 4
21	18, 20	mtbiri 686	. . 3
22		mnfnre 8358	. . . . . . . . . 10
23	22	neli 2517	. . . . . . . . 9
24	23	intnan 941	. . . . . . . 8 $/\$
25	24	intnanr 942	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2466	. . . . . . . 8
27	26	intnan 941	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 825	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 942	. . . . . . 7 $/\$
30	23	intnan 941	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 825	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 825	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8372	. . . . . 6
34		ltxr 10156	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 430	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 682	. . . 4
37		breq12 4130	. . . . 5 $/\$
38	37	anidms 401	. . . 4
39	36, 38	mtbiri 686	. . 3
40	2, 21, 39	3jaoi 1344	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $\/$ w3o 1008

wceq 1402

wcel 2209 class class class wbr 4125

cr 8168

cltrr 8173

cpnf 8347

cmnf 8348

cxr 8349

clt 8350

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-pre-ltirr 8281

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-xp 4775 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355

This theorem is referenced by: xrltnsym 10174 xrltso 10177 xrlttri3 10178 xrleid 10181 xrltne 10194 nltpnft 10195 ngtmnft 10198 xrrebnd 10200 xposdif 10263 lbioog 10294 ubioog 10295 xrmaxleim 11988 xrmaxiflemlub 11992