nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10084

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 8281	. . . . . . 7
2	1	neli 2500	. . . . . 6
3	2	intnan 937	. . . . 5 $/\$
4	3	intnanr 938	. . . 4 $/\$ $/\$
5		pnfnemnf 8293	. . . . . 6
6	5	nesymi 2449	. . . . 5
7	6	intnan 937	. . . 4 $/\$
8	4, 7	pm3.2ni 821	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 937	. . . 4 $/\$
10	2	intnan 937	. . . 4 $/\$
11	9, 10	pm3.2ni 821	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 821	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 8295	. . 3
14		ltxr 10071	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 425	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 682	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2202 class class class wbr 4093

cr 8091

cltrr 8096

cpnf 8270

cmnf 8271

cxr 8272

clt 8273

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278

This theorem is referenced by: mnfle 10088 xrltnsym 10089 xrlttr 10091 xrltso 10092 xltnegi 10131 xposdif 10178 qbtwnxr 10580 xrmaxiflemab 11887 xrmaxltsup 11898 xrbdtri 11916 blssioo 15364