xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9559

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 9556	. 2 $\/$ $\/$
2		ltnr 7834	. . 3
3		pnfnre 7800	. . . . . . . . . 10
4	3	neli 2403	. . . . . . . . 9
5	4	intnan 914	. . . . . . . 8 $/\$
6	5	intnanr 915	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7813	. . . . . . . . 9
8	7	neii 2308	. . . . . . . 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 7811	. . . . . 6
16		ltxr 9555	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 422	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 660	. . . 4
19		breq12 3929	. . . . 5 $/\$
20	19	anidms 394	. . . 4
21	18, 20	mtbiri 664	. . 3
22		mnfnre 7801	. . . . . . . . . 10
23	22	neli 2403	. . . . . . . . 9
24	23	intnan 914	. . . . . . . 8 $/\$
25	24	intnanr 915	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2352	. . . . . . . 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 7815	. . . . . 6
34		ltxr 9555	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 422	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 660	. . . 4
37		breq12 3929	. . . . 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 3924

cr 7612

cltrr 7617

cpnf 7790

cmnf 7791

cxr 7792

clt 7793

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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725

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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798

This theorem is referenced by: xrltnsym 9572 xrltso 9575 xrlttri3 9576 xrleid 9579 xrltne 9589 nltpnft 9590 ngtmnft 9593 xrrebnd 9595 xposdif 9658 lbioog 9689 ubioog 9690 xrmaxleim 11006 xrmaxiflemlub 11010