xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10181

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 10178	. 2 $\/$ $\/$
2		ltnr 8402	. . 3
3		pnfnre 8367	. . . . . . . . . 10
4	3	neli 2517	. . . . . . . . 9
5	4	intnan 941	. . . . . . . 8 $/\$
6	5	intnanr 942	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8380	. . . . . . . . 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 8378	. . . . . 6
16		ltxr 10177	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 430	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 682	. . . 4
19		breq12 4135	. . . . 5 $/\$
20	19	anidms 401	. . . 4
21	18, 20	mtbiri 686	. . 3
22		mnfnre 8368	. . . . . . . . . 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 8382	. . . . . 6
34		ltxr 10177	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 430	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 682	. . . 4
37		breq12 4135	. . . . 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

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

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

wceq 1402

wcel 2209 class class class wbr 4130

cr 8178

cltrr 8183

cpnf 8357

cmnf 8358

cxr 8359

clt 8360

This proof depends on 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 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-pre-ltirr 8291

This proof 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 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-xp 4780 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365

This theorem is used by: xrltnsym 10195 xrltso 10198 xrlttri3 10199 xrleid 10202 xrltne 10215 nltpnft 10216 ngtmnft 10219 xrrebnd 10221 xposdif 10284 lbioog 10315 ubioog 10316 xrmaxleim 12010 xrmaxiflemlub 12014