xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10112

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 10109	. 2 $\/$ $\/$
2		ltnr 8350	. . 3
3		pnfnre 8315	. . . . . . . . . 10
4	3	neli 2509	. . . . . . . . 9
5	4	intnan 937	. . . . . . . 8 $/\$
6	5	intnanr 938	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8328	. . . . . . . . 9
8	7	neii 2414	. . . . . . . 8
9	8	intnanr 938	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 821	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 938	. . . . . . 7 $/\$
12	4	intnan 937	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 821	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 821	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8326	. . . . . 6
16		ltxr 10108	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 678	. . . 4
19		breq12 4114	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 682	. . 3
22		mnfnre 8316	. . . . . . . . . 10
23	22	neli 2509	. . . . . . . . 9
24	23	intnan 937	. . . . . . . 8 $/\$
25	24	intnanr 938	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2458	. . . . . . . 8
27	26	intnan 937	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 821	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 938	. . . . . . 7 $/\$
30	23	intnan 937	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 821	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 821	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8330	. . . . . 6
34		ltxr 10108	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 678	. . . 4
37		breq12 4114	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 682	. . 3
40	2, 21, 39	3jaoi 1340	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $\/$ w3o 1004

wceq 1398

wcel 2203 class class class wbr 4109

cr 8126

cltrr 8131

cpnf 8305

cmnf 8306

cxr 8307

clt 8308

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-pre-ltirr 8239

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-xp 4755 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313

This theorem is referenced by: xrltnsym 10126 xrltso 10129 xrlttri3 10130 xrleid 10133 xrltne 10146 nltpnft 10147 ngtmnft 10150 xrrebnd 10152 xposdif 10215 lbioog 10246 ubioog 10247 xrmaxleim 11929 xrmaxiflemlub 11933