nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10169

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 8358	. . . . . . 7
2	1	neli 2517	. . . . . 6
3	2	intnan 941	. . . . 5 $/\$
4	3	intnanr 942	. . . 4 $/\$ $/\$
5		pnfnemnf 8370	. . . . . 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 8372	. . 3
14		ltxr 10156	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 429	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 686	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209 class class class wbr 4125

cr 8168

cltrr 8173

cpnf 8347

cmnf 8348

cxr 8349

clt 8350

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 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 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261

This theorem 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 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-xp 4775 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355

This theorem is referenced by: mnfle 10173 xrltnsym 10174 xrlttr 10176 xrltso 10177 xltnegi 10216 xposdif 10263 qbtwnxr 10670 xrmaxiflemab 11991 xrmaxltsup 12002 xrbdtri 12020 blssioo 15577