nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 9695

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 7920	. . . . . . 7
2	1	neli 2424	. . . . . 6
3	2	intnan 915	. . . . 5 $/\$
4	3	intnanr 916	. . . 4 $/\$ $/\$
5		pnfnemnf 7932	. . . . . 6
6	5	nesymi 2373	. . . . 5
7	6	intnan 915	. . . 4 $/\$
8	4, 7	pm3.2ni 803	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 915	. . . 4 $/\$
10	2	intnan 915	. . . 4 $/\$
11	9, 10	pm3.2ni 803	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 803	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 7934	. . 3
14		ltxr 9682	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 422	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 665	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1335

wcel 2128 class class class wbr 3965

cr 7731

cltrr 7736

cpnf 7909

cmnf 7910

cxr 7911

clt 7912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917

This theorem is referenced by: mnfle 9699 xrltnsym 9700 xrlttr 9702 xrltso 9703 xltnegi 9739 xposdif 9786 qbtwnxr 10157 xrmaxiflemab 11144 xrmaxltsup 11155 xrbdtri 11173 blssioo 12945