nltmnf - Intuitionistic Logic Explorer

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

Theorem nltmnf 10190

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 8368	. . . . . . 7
2	1	neli 2517	. . . . . 6
3	2	intnan 941	. . . . 5 $/\$
4	3	intnanr 942	. . . 4 $/\$ $/\$
5		pnfnemnf 8380	. . . . . 6
6	5	nesymi 2466	. . . . 5
7	6	intnan 941	. . . 4 $/\$
8	4, 7	pm3.2ni 825	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 941	. . . 4 $/\$
10	2	intnan 941	. . . 4 $/\$
11	9, 10	pm3.2ni 825	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 825	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 8382	. . 3
14		ltxr 10177	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 429	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 686	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209 class class class wbr 4130

cr 8178

cltrr 8183

cpnf 8357

cmnf 8358

cxr 8359

clt 8360

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-xp 4780 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365

This theorem is used by: mnfle 10194 xrltnsym 10195 xrlttr 10197 xrltso 10198 xltnegi 10237 xposdif 10284 qbtwnxr 10692 xrmaxiflemab 12013 xrmaxltsup 12024 xrbdtri 12042 blssioo 15654