xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10058

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 10055	. 2 $\/$ $\/$
2		ltnr 8298	. . 3
3		pnfnre 8263	. . . . . . . . . 10
4	3	neli 2500	. . . . . . . . 9
5	4	intnan 937	. . . . . . . 8 $/\$
6	5	intnanr 938	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8276	. . . . . . . . 9
8	7	neii 2405	. . . . . . . 8
9	8	intnanr 938	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 821	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 938	. . . . . . 7 $/\$
12	4	intnan 937	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 821	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 821	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8274	. . . . . 6
16		ltxr 10054	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 678	. . . 4
19		breq12 4098	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 682	. . 3
22		mnfnre 8264	. . . . . . . . . 10
23	22	neli 2500	. . . . . . . . 9
24	23	intnan 937	. . . . . . . 8 $/\$
25	24	intnanr 938	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2449	. . . . . . . 8
27	26	intnan 937	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 821	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 938	. . . . . . 7 $/\$
30	23	intnan 937	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 821	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 821	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8278	. . . . . 6
34		ltxr 10054	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 678	. . . 4
37		breq12 4098	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 682	. . 3
40	2, 21, 39	3jaoi 1340	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $\/$ w3o 1004

wceq 1398

wcel 2202 class class class wbr 4093

cr 8074

cltrr 8079

cpnf 8253

cmnf 8254

cxr 8255

clt 8256

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-pre-ltirr 8187

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261

This theorem is referenced by: xrltnsym 10072 xrltso 10075 xrlttri3 10076 xrleid 10079 xrltne 10092 nltpnft 10093 ngtmnft 10096 xrrebnd 10098 xposdif 10161 lbioog 10192 ubioog 10193 xrmaxleim 11867 xrmaxiflemlub 11871