ltpnf - Intuitionistic Logic Explorer

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

Theorem ltpnf 10192

Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
ltpnf

**Proof of Theorem ltpnf**
Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4
2		orc 724	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 429	. . 3 $/\$ $\/$ $/\$
4	3	olcd 746	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 8371	. . 3
6		pnfxr 8378	. . 3
7		ltxr 10187	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancl 417	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 167	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209 class class class wbr 4130

cr 8178

cltrr 8183

cpnf 8357

cmnf 8358

cxr 8359

clt 8360

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-cnex 8270

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-xp 4780 df-pnf 8362 df-xr 8364 df-ltxr 8365

This theorem is used by: ltpnfd 10193 0ltpnf 10194 xrlttr 10207 xrltso 10208 xrlttri3 10209 nltpnft 10226 npnflt 10227 xrrebnd 10231 xrre 10232 xltnegi 10247 xltadd1 10288 xposdif 10294 elioc2 10348 elicc2 10350 ioomax 10360 ioopos 10362 elioopnf 10379 elicopnf 10381 qbtwnxr 10702 dfrp2 10708 filtinf 11244 xrmaxltsup 12040 fprodge0 12420 fprodge1 12422 xblss2ps 15554