xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9668

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 9665	. 2 $\/$ $\/$
2		ltnr 7937	. . 3
3		pnfnre 7902	. . . . . . . . . 10
4	3	neli 2424	. . . . . . . . 9
5	4	intnan 915	. . . . . . . 8 $/\$
6	5	intnanr 916	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7915	. . . . . . . . 9
8	7	neii 2329	. . . . . . . 8
9	8	intnanr 916	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 916	. . . . . . 7 $/\$
12	4	intnan 915	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 7913	. . . . . 6
16		ltxr 9664	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 661	. . . 4
19		breq12 3970	. . . . 5 $/\$
20	19	anidms 395	. . . 4
21	18, 20	mtbiri 665	. . 3
22		mnfnre 7903	. . . . . . . . . 10
23	22	neli 2424	. . . . . . . . 9
24	23	intnan 915	. . . . . . . 8 $/\$
25	24	intnanr 916	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2373	. . . . . . . 8
27	26	intnan 915	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 916	. . . . . . 7 $/\$
30	23	intnan 915	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 7917	. . . . . 6
34		ltxr 9664	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 661	. . . 4
37		breq12 3970	. . . . 5 $/\$
38	37	anidms 395	. . . 4
39	36, 38	mtbiri 665	. . 3
40	2, 21, 39	3jaoi 1285	. 2 $\/$ $\/$
41	1, 40	sylbi 120	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 962

wceq 1335

wcel 2128 class class class wbr 3965

cr 7714

cltrr 7719

cpnf 7892

cmnf 7893

cxr 7894

clt 7895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-cnex 7806 ax-resscn 7807 ax-pre-ltirr 7827

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4589 df-pnf 7897 df-mnf 7898 df-xr 7899 df-ltxr 7900

This theorem is referenced by: xrltnsym 9682 xrltso 9685 xrlttri3 9686 xrleid 9689 xrltne 9699 nltpnft 9700 ngtmnft 9703 xrrebnd 9705 xposdif 9768 lbioog 9799 ubioog 9800 xrmaxleim 11123 xrmaxiflemlub 11127