nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10121

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 8316	. . . . . . 7
2	1	neli 2509	. . . . . 6
3	2	intnan 937	. . . . 5 $/\$
4	3	intnanr 938	. . . 4 $/\$ $/\$
5		pnfnemnf 8328	. . . . . 6
6	5	nesymi 2458	. . . . 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 8330	. . 3
14		ltxr 10108	. . 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 2203 class class class wbr 4109

cr 8126

cltrr 8131

cpnf 8305

cmnf 8306

cxr 8307

clt 8308

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 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-xp 4755 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313

This theorem is referenced by: mnfle 10125 xrltnsym 10126 xrlttr 10128 xrltso 10129 xltnegi 10168 xposdif 10215 qbtwnxr 10617 xrmaxiflemab 11932 xrmaxltsup 11943 xrbdtri 11961 blssioo 15418