xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9936

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 9933	. 2 $\/$ $\/$
2		ltnr 8184	. . 3
3		pnfnre 8149	. . . . . . . . . 10
4	3	neli 2475	. . . . . . . . 9
5	4	intnan 931	. . . . . . . 8 $/\$
6	5	intnanr 932	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8162	. . . . . . . . 9
8	7	neii 2380	. . . . . . . 8
9	8	intnanr 932	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 815	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 932	. . . . . . 7 $/\$
12	4	intnan 931	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 815	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 815	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8160	. . . . . 6
16		ltxr 9932	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 673	. . . 4
19		breq12 4064	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 677	. . 3
22		mnfnre 8150	. . . . . . . . . 10
23	22	neli 2475	. . . . . . . . 9
24	23	intnan 931	. . . . . . . 8 $/\$
25	24	intnanr 932	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2424	. . . . . . . 8
27	26	intnan 931	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 815	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 932	. . . . . . 7 $/\$
30	23	intnan 931	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 815	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 815	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8164	. . . . . 6
34		ltxr 9932	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 673	. . . 4
37		breq12 4064	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 677	. . 3
40	2, 21, 39	3jaoi 1316	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 $\/$ w3o 980

wceq 1373

wcel 2178 class class class wbr 4059

cr 7959

cltrr 7964

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-pre-ltirr 8072

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2778 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-xp 4699 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147

This theorem is referenced by: xrltnsym 9950 xrltso 9953 xrlttri3 9954 xrleid 9957 xrltne 9970 nltpnft 9971 ngtmnft 9974 xrrebnd 9976 xposdif 10039 lbioog 10070 ubioog 10071 xrmaxleim 11670 xrmaxiflemlub 11674