ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9775

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 2177	. . . 4
2		orc 712	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 425	. . 3 $/\$ $\/$ $/\$
4	3	olcd 734	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 7998	. . 3
6		pnfxr 8005	. . 3
7		ltxr 9770	. . 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 708

wceq 1353

wcel 2148 class class class wbr 4002

cr 7806

cltrr 7811

cpnf 7984

cmnf 7985

cxr 7986

clt 7987

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7898

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-pnf 7989 df-xr 7991 df-ltxr 7992

This theorem is referenced by: ltpnfd 9776 0ltpnf 9777 xrlttr 9790 xrltso 9791 xrlttri3 9792 nltpnft 9809 npnflt 9810 xrrebnd 9814 xrre 9815 xltnegi 9830 xltadd1 9871 xposdif 9877 elioc2 9931 elicc2 9933 ioomax 9943 ioopos 9945 elioopnf 9962 elicopnf 9964 qbtwnxr 10252 dfrp2 10258 filtinf 10763 xrmaxltsup 11258 fprodge0 11637 fprodge1 11639 xblss2ps 13766