xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9596

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 9593	. 2 $\/$ $\/$
2		ltnr 7865	. . 3
3		pnfnre 7831	. . . . . . . . . 10
4	3	neli 2406	. . . . . . . . 9
5	4	intnan 915	. . . . . . . 8 $/\$
6	5	intnanr 916	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7844	. . . . . . . . 9
8	7	neii 2311	. . . . . . . 8
9	8	intnanr 916	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 916	. . . . . . 7 $/\$
12	4	intnan 915	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 7842	. . . . . 6
16		ltxr 9592	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 661	. . . 4
19		breq12 3942	. . . . 5 $/\$
20	19	anidms 395	. . . 4
21	18, 20	mtbiri 665	. . 3
22		mnfnre 7832	. . . . . . . . . 10
23	22	neli 2406	. . . . . . . . 9
24	23	intnan 915	. . . . . . . 8 $/\$
25	24	intnanr 916	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2355	. . . . . . . 8
27	26	intnan 915	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 916	. . . . . . 7 $/\$
30	23	intnan 915	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 7846	. . . . . 6
34		ltxr 9592	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 661	. . . 4
37		breq12 3942	. . . . 5 $/\$
38	37	anidms 395	. . . 4
39	36, 38	mtbiri 665	. . 3
40	2, 21, 39	3jaoi 1282	. 2 $\/$ $\/$
41	1, 40	sylbi 120	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 962

wceq 1332

wcel 1481 class class class wbr 3937

cr 7643

cltrr 7648

cpnf 7821

cmnf 7822

cxr 7823

clt 7824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829

This theorem is referenced by: xrltnsym 9609 xrltso 9612 xrlttri3 9613 xrleid 9616 xrltne 9626 nltpnft 9627 ngtmnft 9630 xrrebnd 9632 xposdif 9695 lbioog 9726 ubioog 9727 xrmaxleim 11045 xrmaxiflemlub 11049