xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9736

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 9733	. 2 $\/$ $\/$
2		ltnr 7996	. . 3
3		pnfnre 7961	. . . . . . . . . 10
4	3	neli 2437	. . . . . . . . 9
5	4	intnan 924	. . . . . . . 8 $/\$
6	5	intnanr 925	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7974	. . . . . . . . 9
8	7	neii 2342	. . . . . . . 8
9	8	intnanr 925	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 808	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 925	. . . . . . 7 $/\$
12	4	intnan 924	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 808	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 808	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 7972	. . . . . 6
16		ltxr 9732	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 424	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 666	. . . 4
19		breq12 3994	. . . . 5 $/\$
20	19	anidms 395	. . . 4
21	18, 20	mtbiri 670	. . 3
22		mnfnre 7962	. . . . . . . . . 10
23	22	neli 2437	. . . . . . . . 9
24	23	intnan 924	. . . . . . . 8 $/\$
25	24	intnanr 925	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2386	. . . . . . . 8
27	26	intnan 924	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 808	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 925	. . . . . . 7 $/\$
30	23	intnan 924	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 808	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 808	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 7976	. . . . . 6
34		ltxr 9732	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 424	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 666	. . . 4
37		breq12 3994	. . . . 5 $/\$
38	37	anidms 395	. . . 4
39	36, 38	mtbiri 670	. . 3
40	2, 21, 39	3jaoi 1298	. 2 $\/$ $\/$
41	1, 40	sylbi 120	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $\/$ w3o 972

wceq 1348

wcel 2141 class class class wbr 3989

cr 7773

cltrr 7778

cpnf 7951

cmnf 7952

cxr 7953

clt 7954

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959

This theorem is referenced by: xrltnsym 9750 xrltso 9753 xrlttri3 9754 xrleid 9757 xrltne 9770 nltpnft 9771 ngtmnft 9774 xrrebnd 9776 xposdif 9839 lbioog 9870 ubioog 9871 xrmaxleim 11207 xrmaxiflemlub 11211