xrltnr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrltnr		Unicode version

Theorem xrltnr 9900

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 9897	. 2 $\/$ $\/$
2		ltnr 8148	. . 3
3		pnfnre 8113	. . . . . . . . . 10
4	3	neli 2472	. . . . . . . . 9
5	4	intnan 930	. . . . . . . 8 $/\$
6	5	intnanr 931	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8126	. . . . . . . . 9
8	7	neii 2377	. . . . . . . 8
9	8	intnanr 931	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 814	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 931	. . . . . . 7 $/\$
12	4	intnan 930	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 814	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 814	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8124	. . . . . 6
16		ltxr 9896	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 672	. . . 4
19		breq12 4048	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 676	. . 3
22		mnfnre 8114	. . . . . . . . . 10
23	22	neli 2472	. . . . . . . . 9
24	23	intnan 930	. . . . . . . 8 $/\$
25	24	intnanr 931	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2421	. . . . . . . 8
27	26	intnan 930	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 814	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 931	. . . . . . 7 $/\$
30	23	intnan 930	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 814	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 814	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8128	. . . . . 6
34		ltxr 9896	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 672	. . . 4
37		breq12 4048	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 676	. . 3
40	2, 21, 39	3jaoi 1315	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $\/$ w3o 979

wceq 1372

wcel 2175 class class class wbr 4043

cr 7923

cltrr 7928

cpnf 8103

cmnf 8104

cxr 8105

clt 8106

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-pre-ltirr 8036

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-xp 4680 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111

This theorem is referenced by: xrltnsym 9914 xrltso 9917 xrlttri3 9918 xrleid 9921 xrltne 9934 nltpnft 9935 ngtmnft 9938 xrrebnd 9940 xposdif 10003 lbioog 10034 ubioog 10035 xrmaxleim 11526 xrmaxiflemlub 11530