xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 10131

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 10128	. 2 $\/$ $\/$
2		ltnr 8366	. . 3
3		pnfnre 8331	. . . . . . . . . 10
4	3	neli 2511	. . . . . . . . 9
5	4	intnan 937	. . . . . . . 8 $/\$
6	5	intnanr 938	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8344	. . . . . . . . 9
8	7	neii 2416	. . . . . . . 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 8342	. . . . . 6
16		ltxr 10127	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 678	. . . 4
19		breq12 4119	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 682	. . 3
22		mnfnre 8332	. . . . . . . . . 10
23	22	neli 2511	. . . . . . . . 9
24	23	intnan 937	. . . . . . . 8 $/\$
25	24	intnanr 938	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2460	. . . . . . . 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 8346	. . . . . 6
34		ltxr 10127	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 678	. . . 4
37		breq12 4119	. . . . 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 2205 class class class wbr 4114

cr 8142

cltrr 8147

cpnf 8321

cmnf 8322

cxr 8323

clt 8324

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-pre-ltirr 8255

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 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-xp 4760 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329

This theorem is referenced by: xrltnsym 10145 xrltso 10148 xrlttri3 10149 xrleid 10152 xrltne 10165 nltpnft 10166 ngtmnft 10169 xrrebnd 10171 xposdif 10234 lbioog 10265 ubioog 10266 xrmaxleim 11954 xrmaxiflemlub 11958