xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9774

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 9771	. 2 $\/$ $\/$
2		ltnr 8029	. . 3
3		pnfnre 7994	. . . . . . . . . 10
4	3	neli 2444	. . . . . . . . 9
5	4	intnan 929	. . . . . . . 8 $/\$
6	5	intnanr 930	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 8007	. . . . . . . . 9
8	7	neii 2349	. . . . . . . 8
9	8	intnanr 930	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 813	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 930	. . . . . . 7 $/\$
12	4	intnan 929	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 813	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 813	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 8005	. . . . . 6
16		ltxr 9770	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 671	. . . 4
19		breq12 4007	. . . . 5 $/\$
20	19	anidms 397	. . . 4
21	18, 20	mtbiri 675	. . 3
22		mnfnre 7995	. . . . . . . . . 10
23	22	neli 2444	. . . . . . . . 9
24	23	intnan 929	. . . . . . . 8 $/\$
25	24	intnanr 930	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2393	. . . . . . . 8
27	26	intnan 929	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 813	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 930	. . . . . . 7 $/\$
30	23	intnan 929	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 813	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 813	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 8009	. . . . . 6
34		ltxr 9770	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 426	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 671	. . . 4
37		breq12 4007	. . . . 5 $/\$
38	37	anidms 397	. . . 4
39	36, 38	mtbiri 675	. . 3
40	2, 21, 39	3jaoi 1303	. 2 $\/$ $\/$
41	1, 40	sylbi 121	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 $\/$ w3o 977

wceq 1353

wcel 2148 class class class wbr 4002

cr 7806

cltrr 7811

cpnf 7984

cmnf 7985

cxr 7986

clt 7987

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-pre-ltirr 7919

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992

This theorem is referenced by: xrltnsym 9788 xrltso 9791 xrlttri3 9792 xrleid 9795 xrltne 9808 nltpnft 9809 ngtmnft 9812 xrrebnd 9814 xposdif 9877 lbioog 9908 ubioog 9909 xrmaxleim 11244 xrmaxiflemlub 11248