nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 9567

Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
nltmnf

**Proof of Theorem nltmnf**
Step	Hyp	Ref	Expression
1		mnfnre 7801	. . . . . . 7
2	1	neli 2403	. . . . . 6
3	2	intnan 914	. . . . 5 $/\$
4	3	intnanr 915	. . . 4 $/\$ $/\$
5		pnfnemnf 7813	. . . . . 6
6	5	nesymi 2352	. . . . 5
7	6	intnan 914	. . . 4 $/\$
8	4, 7	pm3.2ni 802	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 914	. . . 4 $/\$
10	2	intnan 914	. . . 4 $/\$
11	9, 10	pm3.2ni 802	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 802	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 7815	. . 3
14		ltxr 9555	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 421	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 664	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697

wceq 1331

wcel 1480 class class class wbr 3924

cr 7612

cltrr 7617

cpnf 7790

cmnf 7791

cxr 7792

clt 7793

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798

This theorem is referenced by: mnfle 9571 xrltnsym 9572 xrlttr 9574 xrltso 9575 xltnegi 9611 xposdif 9658 qbtwnxr 10028 xrmaxiflemab 11009 xrmaxltsup 11020 xrbdtri 11038 blssioo 12703