xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9566

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 9563	. 2 $\/$ $\/$
2		ltnr 7841	. . 3
3		pnfnre 7807	. . . . . . . . . 10
4	3	neli 2405	. . . . . . . . 9
5	4	intnan 914	. . . . . . . 8 $/\$
6	5	intnanr 915	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7820	. . . . . . . . 9
8	7	neii 2310	. . . . . . . 8
9	8	intnanr 915	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 802	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 915	. . . . . . 7 $/\$
12	4	intnan 914	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 802	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 802	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 7818	. . . . . 6
16		ltxr 9562	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 422	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 660	. . . 4
19		breq12 3934	. . . . 5 $/\$
20	19	anidms 394	. . . 4
21	18, 20	mtbiri 664	. . 3
22		mnfnre 7808	. . . . . . . . . 10
23	22	neli 2405	. . . . . . . . 9
24	23	intnan 914	. . . . . . . 8 $/\$
25	24	intnanr 915	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2354	. . . . . . . 8
27	26	intnan 914	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 802	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 915	. . . . . . 7 $/\$
30	23	intnan 914	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 802	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 802	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 7822	. . . . . 6
34		ltxr 9562	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 422	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 660	. . . 4
37		breq12 3934	. . . . 5 $/\$
38	37	anidms 394	. . . 4
39	36, 38	mtbiri 664	. . 3
40	2, 21, 39	3jaoi 1281	. 2 $\/$ $\/$
41	1, 40	sylbi 120	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697 $\/$ w3o 961

wceq 1331

wcel 1480 class class class wbr 3929

cr 7619

cltrr 7624

cpnf 7797

cmnf 7798

cxr 7799

clt 7800

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805

This theorem is referenced by: xrltnsym 9579 xrltso 9582 xrlttri3 9583 xrleid 9586 xrltne 9596 nltpnft 9597 ngtmnft 9600 xrrebnd 9602 xposdif 9665 lbioog 9696 ubioog 9697 xrmaxleim 11013 xrmaxiflemlub 11017