xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 8855

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 8850	. 2 $\/$ $\/$
2		ltnr 7188	. . 3
3		pnfnre 7160	. . . . . . . . . 10
4	3	neli 2341	. . . . . . . . 9
5	4	intnan 871	. . . . . . . 8 $/\$
6	5	intnanr 872	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8851	. . . . . . . . 9
8	7	neii 2247	. . . . . . . 8
9	8	intnanr 872	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 759	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 872	. . . . . . 7 $/\$
12	4	intnan 871	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 759	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 759	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8846	. . . . . 6
16		ltxr 8849	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 416	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 628	. . . 4
19		breq12 3790	. . . . 5 $/\$
20	19	anidms 389	. . . 4
21	18, 20	mtbiri 632	. . 3
22		mnfnre 7161	. . . . . . . . . 10
23	22	neli 2341	. . . . . . . . 9
24	23	intnan 871	. . . . . . . 8 $/\$
25	24	intnanr 872	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2291	. . . . . . . 8
27	26	intnan 871	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 759	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 872	. . . . . . 7 $/\$
30	23	intnan 871	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 759	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 759	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8848	. . . . . 6
34		ltxr 8849	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 416	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 628	. . . 4
37		breq12 3790	. . . . 5 $/\$
38	37	anidms 389	. . . 4
39	36, 38	mtbiri 632	. . 3
40	2, 21, 39	3jaoi 1234	. 2 $\/$ $\/$
41	1, 40	sylbi 119	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661 $\/$ w3o 918

wceq 1284

wcel 1433 class class class wbr 3785

cr 6980

cltrr 6985

cpnf 7150

cmnf 7151

cxr 7152

clt 7153

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158

This theorem is referenced by: xrltnsym 8868 xrltso 8871 xrlttri3 8872 xrleid 8874 xrltne 8883 nltpnft 8884 ngtmnft 8885 xrrebnd 8886 lbioog 8936 ubioog 8937