nltmnf - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > nltmnf		Unicode version

Theorem nltmnf 9784

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 7996	. . . . . . 7
2	1	neli 2444	. . . . . 6
3	2	intnan 929	. . . . 5 $/\$
4	3	intnanr 930	. . . 4 $/\$ $/\$
5		pnfnemnf 8008	. . . . . 6
6	5	nesymi 2393	. . . . 5
7	6	intnan 929	. . . 4 $/\$
8	4, 7	pm3.2ni 813	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 929	. . . 4 $/\$
10	2	intnan 929	. . . 4 $/\$
11	9, 10	pm3.2ni 813	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 813	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 8010	. . 3
14		ltxr 9771	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 425	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 675	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wceq 1353

wcel 2148 class class class wbr 4002

cr 7807

cltrr 7812

cpnf 7985

cmnf 7986

cxr 7987

clt 7988

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 7899 ax-resscn 7900

This theorem depends on definitions: df-bi 117 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 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993

This theorem is referenced by: mnfle 9788 xrltnsym 9789 xrlttr 9791 xrltso 9792 xltnegi 9831 xposdif 9878 qbtwnxr 10253 xrmaxiflemab 11248 xrmaxltsup 11259 xrbdtri 11277 blssioo 13916