xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9854

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 9851	. 2 $\/$ $\/$
2		ltnr 8103	. . 3
3		pnfnre 8068	. . . . . . . . . 10
4	3	neli 2464	. . . . . . . . 9
5	4	intnan 930	. . . . . . . 8 $/\$
6	5	intnanr 931	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8081	. . . . . . . . 9
8	7	neii 2369	. . . . . . . 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 8079	. . . . . 6
16		ltxr 9850	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 672	. . . 4
19		breq12 4038	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 676	. . 3
22		mnfnre 8069	. . . . . . . . . 10
23	22	neli 2464	. . . . . . . . 9
24	23	intnan 930	. . . . . . . 8 $/\$
25	24	intnanr 931	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2413	. . . . . . . 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 8083	. . . . . 6
34		ltxr 9850	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 672	. . . 4
37		breq12 4038	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 676	. . 3
40	2, 21, 39	3jaoi 1314	. 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 1364

wcel 2167 class class class wbr 4033

cr 7878

cltrr 7883

cpnf 8058

cmnf 8059

cxr 8060

clt 8061

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-pre-ltirr 7991

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-xp 4669 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066

This theorem is referenced by: xrltnsym 9868 xrltso 9871 xrlttri3 9872 xrleid 9875 xrltne 9888 nltpnft 9889 ngtmnft 9892 xrrebnd 9894 xposdif 9957 lbioog 9988 ubioog 9989 xrmaxleim 11409 xrmaxiflemlub 11413