nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10140

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 8332	. . . . . . 7
2	1	neli 2511	. . . . . 6
3	2	intnan 937	. . . . 5 $/\$
4	3	intnanr 938	. . . 4 $/\$ $/\$
5		pnfnemnf 8344	. . . . . 6
6	5	nesymi 2460	. . . . 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 8346	. . 3
14		ltxr 10127	. . 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 2205 class class class wbr 4114

cr 8142

cltrr 8147

cpnf 8321

cmnf 8322

cxr 8323

clt 8324

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-xp 4760 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329

This theorem is referenced by: mnfle 10144 xrltnsym 10145 xrlttr 10147 xrltso 10148 xltnegi 10187 xposdif 10234 qbtwnxr 10641 xrmaxiflemab 11957 xrmaxltsup 11968 xrbdtri 11986 blssioo 15544