ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 10076

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 2231	. . . 4
2		orc 720	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 425	. . 3 $/\$ $\/$ $/\$
4	3	olcd 742	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 8284	. . 3
6		pnfxr 8291	. . 3
7		ltxr 10071	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancl 413	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 167	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2202 class class class wbr 4093

cr 8091

cltrr 8096

cpnf 8270

cmnf 8271

cxr 8272

clt 8273

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-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-cnex 8183

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-pnf 8275 df-xr 8277 df-ltxr 8278

This theorem is referenced by: ltpnfd 10077 0ltpnf 10078 xrlttr 10091 xrltso 10092 xrlttri3 10093 nltpnft 10110 npnflt 10111 xrrebnd 10115 xrre 10116 xltnegi 10131 xltadd1 10172 xposdif 10178 elioc2 10232 elicc2 10234 ioomax 10244 ioopos 10246 elioopnf 10263 elicopnf 10265 qbtwnxr 10580 dfrp2 10586 filtinf 11116 xrmaxltsup 11898 fprodge0 12278 fprodge1 12280 xblss2ps 15215