nltmnf - Intuitionistic Logic Explorer

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Theorem nltmnf 9461

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 7726	. . . . . . 7
2	1	neli 2377	. . . . . 6
3	2	intnan 895	. . . . 5 $/\$
4	3	intnanr 896	. . . 4 $/\$ $/\$
5		pnfnemnf 7738	. . . . . 6
6	5	nesymi 2326	. . . . 5
7	6	intnan 895	. . . 4 $/\$
8	4, 7	pm3.2ni 785	. . 3 $/\$ $/\$ $\/$ $/\$
9	6	intnan 895	. . . 4 $/\$
10	2	intnan 895	. . . 4 $/\$
11	9, 10	pm3.2ni 785	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 785	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		mnfxr 7740	. . 3
14		ltxr 9449	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan2 419	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 647	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 680

wceq 1312

wcel 1461 class class class wbr 3893

cr 7540

cltrr 7545

cpnf 7715

cmnf 7716

cxr 7717

clt 7718

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631

This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-xp 4503 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723

This theorem is referenced by: mnfle 9465 xrltnsym 9466 xrlttr 9468 xrltso 9469 xltnegi 9505 xposdif 9552 qbtwnxr 9922 xrmaxiflemab 10902 xrmaxltsup 10913 xrbdtri 10931 blssioo 12525