nltmnf - Intuitionistic Logic Explorer

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

Theorem nltmnf 9910

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 8115	. . . . . . 7
2	1	neli 2473	. . . . . 6
3	2	intnan 931	. . . . 5 $/\$
4	3	intnanr 932	. . . 4 $/\$ $/\$
5		pnfnemnf 8127	. . . . . 6
6	5	nesymi 2422	. . . . 5
7	6	intnan 931	. . . 4 $/\$
8	4, 7	pm3.2ni 815	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 931	. . . 4 $/\$
10	2	intnan 931	. . . 4 $/\$
11	9, 10	pm3.2ni 815	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 815	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 8129	. . 3
14		ltxr 9897	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 425	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 677	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710

wceq 1373

wcel 2176 class class class wbr 4044

cr 7924

cltrr 7929

cpnf 8104

cmnf 8105

cxr 8106

clt 8107

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-xp 4681 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112

This theorem is referenced by: mnfle 9914 xrltnsym 9915 xrlttr 9917 xrltso 9918 xltnegi 9957 xposdif 10004 qbtwnxr 10400 xrmaxiflemab 11558 xrmaxltsup 11569 xrbdtri 11587 blssioo 15025