xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9848

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 9845	. 2 $\/$ $\/$
2		ltnr 8098	. . 3
3		pnfnre 8063	. . . . . . . . . 10
4	3	neli 2461	. . . . . . . . 9
5	4	intnan 930	. . . . . . . 8 $/\$
6	5	intnanr 931	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8076	. . . . . . . . 9
8	7	neii 2366	. . . . . . . 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 8074	. . . . . 6
16		ltxr 9844	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 672	. . . 4
19		breq12 4035	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 676	. . 3
22		mnfnre 8064	. . . . . . . . . 10
23	22	neli 2461	. . . . . . . . 9
24	23	intnan 930	. . . . . . . 8 $/\$
25	24	intnanr 931	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2410	. . . . . . . 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 8078	. . . . . 6
34		ltxr 9844	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 672	. . . 4
37		breq12 4035	. . . . 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 2164 class class class wbr 4030

cr 7873

cltrr 7878

cpnf 8053

cmnf 8054

cxr 8055

clt 8056

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-pre-ltirr 7986

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061

This theorem is referenced by: xrltnsym 9862 xrltso 9865 xrlttri3 9866 xrleid 9869 xrltne 9882 nltpnft 9883 ngtmnft 9886 xrrebnd 9888 xposdif 9951 lbioog 9982 ubioog 9983 xrmaxleim 11390 xrmaxiflemlub 11394