ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9872

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 2196	. . . 4
2		orc 713	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 425	. . 3 $/\$ $\/$ $/\$
4	3	olcd 735	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 8089	. . 3
6		pnfxr 8096	. . 3
7		ltxr 9867	. . 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 709

wceq 1364

wcel 2167 class class class wbr 4034

cr 7895

cltrr 7900

cpnf 8075

cmnf 8076

cxr 8077

clt 8078

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-cnex 7987

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-xp 4670 df-pnf 8080 df-xr 8082 df-ltxr 8083

This theorem is referenced by: ltpnfd 9873 0ltpnf 9874 xrlttr 9887 xrltso 9888 xrlttri3 9889 nltpnft 9906 npnflt 9907 xrrebnd 9911 xrre 9912 xltnegi 9927 xltadd1 9968 xposdif 9974 elioc2 10028 elicc2 10030 ioomax 10040 ioopos 10042 elioopnf 10059 elicopnf 10061 qbtwnxr 10364 dfrp2 10370 filtinf 10900 xrmaxltsup 11440 fprodge0 11819 fprodge1 11821 xblss2ps 14724