xrltnr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrltnr		Unicode version

Theorem xrltnr 9845

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 9842	. 2 $\/$ $\/$
2		ltnr 8096	. . 3
3		pnfnre 8061	. . . . . . . . . 10
4	3	neli 2461	. . . . . . . . 9
5	4	intnan 930	. . . . . . . 8 $/\$
6	5	intnanr 931	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8074	. . . . . . . . 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 8072	. . . . . 6
16		ltxr 9841	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 672	. . . 4
19		breq12 4034	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 676	. . 3
22		mnfnre 8062	. . . . . . . . . 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 8076	. . . . . 6
34		ltxr 9841	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 672	. . . 4
37		breq12 4034	. . . . 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 4029

cr 7871

cltrr 7876

cpnf 8051

cmnf 8052

cxr 8053

clt 8054

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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-pre-ltirr 7984

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 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059

This theorem is referenced by: xrltnsym 9859 xrltso 9862 xrlttri3 9863 xrleid 9866 xrltne 9879 nltpnft 9880 ngtmnft 9883 xrrebnd 9885 xposdif 9948 lbioog 9979 ubioog 9980 xrmaxleim 11387 xrmaxiflemlub 11391