xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10013

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 10010	. 2 $\/$ $\/$
2		ltnr 8255	. . 3
3		pnfnre 8220	. . . . . . . . . 10
4	3	neli 2499	. . . . . . . . 9
5	4	intnan 936	. . . . . . . 8 $/\$
6	5	intnanr 937	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8233	. . . . . . . . 9
8	7	neii 2404	. . . . . . . 8
9	8	intnanr 937	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 820	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 937	. . . . . . 7 $/\$
12	4	intnan 936	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 820	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 820	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8231	. . . . . 6
16		ltxr 10009	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 677	. . . 4
19		breq12 4093	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 681	. . 3
22		mnfnre 8221	. . . . . . . . . 10
23	22	neli 2499	. . . . . . . . 9
24	23	intnan 936	. . . . . . . 8 $/\$
25	24	intnanr 937	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2448	. . . . . . . 8
27	26	intnan 936	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 820	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 937	. . . . . . 7 $/\$
30	23	intnan 936	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 820	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 820	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8235	. . . . . 6
34		ltxr 10009	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 677	. . . 4
37		breq12 4093	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 681	. . 3
40	2, 21, 39	3jaoi 1339	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 715 $\/$ w3o 1003

wceq 1397

wcel 2202 class class class wbr 4088

cr 8030

cltrr 8035

cpnf 8210

cmnf 8211

cxr 8212

clt 8213

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-pre-ltirr 8143

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-xp 4731 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218

This theorem is referenced by: xrltnsym 10027 xrltso 10030 xrlttri3 10031 xrleid 10034 xrltne 10047 nltpnft 10048 ngtmnft 10051 xrrebnd 10053 xposdif 10116 lbioog 10147 ubioog 10148 xrmaxleim 11804 xrmaxiflemlub 11808