xrltnr - Intuitionistic Logic Explorer

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Theorem xrltnr 9715

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 9712	. 2 $\/$ $\/$
2		ltnr 7975	. . 3
3		pnfnre 7940	. . . . . . . . . 10
4	3	neli 2433	. . . . . . . . 9
5	4	intnan 919	. . . . . . . 8 $/\$
6	5	intnanr 920	. . . . . . 7 $/\$ $/\$
7		pnfnemnf 7953	. . . . . . . . 9
8	7	neii 2338	. . . . . . . 8
9	8	intnanr 920	. . . . . . 7 $/\$
10	6, 9	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
11	4	intnanr 920	. . . . . . 7 $/\$
12	4	intnan 919	. . . . . . 7 $/\$
13	11, 12	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
14	10, 13	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15		pnfxr 7951	. . . . . 6
16		ltxr 9711	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
17	15, 15, 16	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
18	14, 17	mtbir 661	. . . 4
19		breq12 3987	. . . . 5 $/\$
20	19	anidms 395	. . . 4
21	18, 20	mtbiri 665	. . 3
22		mnfnre 7941	. . . . . . . . . 10
23	22	neli 2433	. . . . . . . . 9
24	23	intnan 919	. . . . . . . 8 $/\$
25	24	intnanr 920	. . . . . . 7 $/\$ $/\$
26	7	nesymi 2382	. . . . . . . 8
27	26	intnan 919	. . . . . . 7 $/\$
28	25, 27	pm3.2ni 803	. . . . . 6 $/\$ $/\$ $\/$ $/\$
29	23	intnanr 920	. . . . . . 7 $/\$
30	23	intnan 919	. . . . . . 7 $/\$
31	29, 30	pm3.2ni 803	. . . . . 6 $/\$ $\/$ $/\$
32	28, 31	pm3.2ni 803	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		mnfxr 7955	. . . . . 6
34		ltxr 9711	. . . . . 6 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
35	33, 33, 34	mp2an 423	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	32, 35	mtbir 661	. . . 4
37		breq12 3987	. . . . 5 $/\$
38	37	anidms 395	. . . 4
39	36, 38	mtbiri 665	. . 3
40	2, 21, 39	3jaoi 1293	. 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 967

wceq 1343

wcel 2136 class class class wbr 3982

cr 7752

cltrr 7757

cpnf 7930

cmnf 7931

cxr 7932

clt 7933

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-pre-ltirr 7865

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938

This theorem is referenced by: xrltnsym 9729 xrltso 9732 xrlttri3 9733 xrleid 9736 xrltne 9749 nltpnft 9750 ngtmnft 9753 xrrebnd 9755 xposdif 9818 lbioog 9849 ubioog 9850 xrmaxleim 11185 xrmaxiflemlub 11189