pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 10012

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 8211	. . . . . . 7
2	1	neli 2497	. . . . . 6
3	2	intnanr 935	. . . . 5 $/\$
4	3	intnanr 935	. . . 4 $/\$ $/\$
5		pnfnemnf 8224	. . . . . 6
6	5	neii 2402	. . . . 5
7	6	intnanr 935	. . . 4 $/\$
8	4, 7	pm3.2ni 818	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 935	. . . 4 $/\$
10	6	intnanr 935	. . . 4 $/\$
11	9, 10	pm3.2ni 818	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 818	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 8222	. . 3
14		ltxr 10000	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 424	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 679	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 713

wceq 1395

wcel 2200 class class class wbr 4086

cr 8021

cltrr 8026

cpnf 8201

cmnf 8202

cxr 8203

clt 8204

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-cnex 8113 ax-resscn 8114

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2802 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-xp 4729 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209

This theorem is referenced by: pnfge 10014 xrltnsym 10018 xrlttr 10020 xrltso 10021 xltnegi 10060 xposdif 10107 qbtwnxr 10507 xqltnle 10517 xrmaxiflemab 11798 xrmaxltsup 11809