pnfnlt - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pnfnlt		Unicode version

Theorem pnfnlt 9573

Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
pnfnlt

**Proof of Theorem pnfnlt**
Step	Hyp	Ref	Expression
1		pnfnre 7807	. . . . . . 7
2	1	neli 2405	. . . . . 6
3	2	intnanr 915	. . . . 5 $/\$
4	3	intnanr 915	. . . 4 $/\$ $/\$
5		pnfnemnf 7820	. . . . . 6
6	5	neii 2310	. . . . 5
7	6	intnanr 915	. . . 4 $/\$
8	4, 7	pm3.2ni 802	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 915	. . . 4 $/\$
10	6	intnanr 915	. . . 4 $/\$
11	9, 10	pm3.2ni 802	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 802	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 7818	. . 3
14		ltxr 9562	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 420	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 664	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697

wceq 1331

wcel 1480 class class class wbr 3929

cr 7619

cltrr 7624

cpnf 7797

cmnf 7798

cxr 7799

clt 7800

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805

This theorem is referenced by: pnfge 9575 xrltnsym 9579 xrlttr 9581 xrltso 9582 xltnegi 9618 xposdif 9665 qbtwnxr 10035 xrmaxiflemab 11016 xrmaxltsup 11027