xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9901

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 9898	. 2 $\/$ $\/$
2		ltnr 8149	. . 3
3		pnfnre 8114	. . . . . . . . . 10
4	3	neli 2473	. . . . . . . . 9
5	4	intnan 931	. . . . . . . 8 $/\$
6	5	intnanr 932	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8127	. . . . . . . . 9
8	7	neii 2378	. . . . . . . 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 8125	. . . . . 6
16		ltxr 9897	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 673	. . . 4
19		breq12 4049	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 677	. . 3
22		mnfnre 8115	. . . . . . . . . 10
23	22	neli 2473	. . . . . . . . 9
24	23	intnan 931	. . . . . . . 8 $/\$
25	24	intnanr 932	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2422	. . . . . . . 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 8129	. . . . . 6
34		ltxr 9897	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 673	. . . 4
37		breq12 4049	. . . . 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 2176 class class class wbr 4044

cr 7924

cltrr 7929

cpnf 8104

cmnf 8105

cxr 8106

clt 8107

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-pre-ltirr 8037

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-xp 4681 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112

This theorem is referenced by: xrltnsym 9915 xrltso 9918 xrlttri3 9919 xrleid 9922 xrltne 9935 nltpnft 9936 ngtmnft 9939 xrrebnd 9941 xposdif 10004 lbioog 10035 ubioog 10036 xrmaxleim 11555 xrmaxiflemlub 11559