nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10144

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 8333	. . . . . . 7
2	1	neli 2511	. . . . . 6
3	2	intnan 937	. . . . 5 $/\$
4	3	intnanr 938	. . . 4 $/\$ $/\$
5		pnfnemnf 8345	. . . . . 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 8347	. . 3
14		ltxr 10131	. . 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 4115

cr 8143

cltrr 8148

cpnf 8322

cmnf 8323

cxr 8324

clt 8325

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 4234 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4665 ax-cnex 8235 ax-resscn 8236

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 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-br 4116 df-opab 4178 df-xp 4761 df-pnf 8327 df-mnf 8328 df-xr 8329 df-ltxr 8330

This theorem is referenced by: mnfle 10148 xrltnsym 10149 xrlttr 10151 xrltso 10152 xltnegi 10191 xposdif 10238 qbtwnxr 10645 xrmaxiflemab 11962 xrmaxltsup 11973 xrbdtri 11991 blssioo 15549