pnfnlt - Metamath Proof Explorer

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Theorem pnfnlt 11962

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 10081	. . . . . . 7
2	1	neli 2899	. . . . . 6
3	2	intnanr 961	. . . . 5 $/\$
4	3	intnanr 961	. . . 4 $/\$ $/\$
5		pnfnemnf 10094	. . . . . 6
6	5	neii 2796	. . . . 5
7	6	intnanr 961	. . . 4 $/\$
8	4, 7	pm3.2ni 899	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 961	. . . 4 $/\$
10	6	intnanr 961	. . . 4 $/\$
11	9, 10	pm3.2ni 899	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 899	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 10092	. . 3
14		ltxr 11949	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 706	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 317	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990 class class class wbr 4653

cr 9935

cltrr 9940

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079

This theorem is referenced by: pnfge 11964 xrltnsym 11970 xrlttr 11973 qbtwnxr 12031 xltnegi 12047 xmullem2 12095 xrinfmexpnf 12136 xrsupsslem 12137 xrinfmsslem 12138 xrub 12142 supxrpnf 12148 supxrunb1 12149 supxrunb2 12150 xrinf0 12168 lt6abl 18296 pnfnei 21024 metdstri 22654 esumpcvgval 30140 icorempt2 33199 iooelexlt 33210 iccpartigtl 41359