nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 10022

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 8221	. . . . . . 7
2	1	neli 2499	. . . . . 6
3	2	intnan 936	. . . . 5 $/\$
4	3	intnanr 937	. . . 4 $/\$ $/\$
5		pnfnemnf 8233	. . . . . 6
6	5	nesymi 2448	. . . . 5
7	6	intnan 936	. . . 4 $/\$
8	4, 7	pm3.2ni 820	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 936	. . . 4 $/\$
10	2	intnan 936	. . . 4 $/\$
11	9, 10	pm3.2ni 820	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 820	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 8235	. . 3
14		ltxr 10009	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 425	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 681	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 715

wceq 1397

wcel 2202 class class class wbr 4088

cr 8030

cltrr 8035

cpnf 8210

cmnf 8211

cxr 8212

clt 8213

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-xp 4731 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218

This theorem is referenced by: mnfle 10026 xrltnsym 10027 xrlttr 10029 xrltso 10030 xltnegi 10069 xposdif 10116 qbtwnxr 10516 xrmaxiflemab 11807 xrmaxltsup 11818 xrbdtri 11836 blssioo 15276