ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9937

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 2207	. . . 4
2		orc 714	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 425	. . 3 $/\$ $\/$ $/\$
4	3	olcd 736	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 8153	. . 3
6		pnfxr 8160	. . 3
7		ltxr 9932	. . 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 710

wceq 1373

wcel 2178 class class class wbr 4059

cr 7959

cltrr 7964

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-cnex 8051

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-xp 4699 df-pnf 8144 df-xr 8146 df-ltxr 8147

This theorem is referenced by: ltpnfd 9938 0ltpnf 9939 xrlttr 9952 xrltso 9953 xrlttri3 9954 nltpnft 9971 npnflt 9972 xrrebnd 9976 xrre 9977 xltnegi 9992 xltadd1 10033 xposdif 10039 elioc2 10093 elicc2 10095 ioomax 10105 ioopos 10107 elioopnf 10124 elicopnf 10126 qbtwnxr 10437 dfrp2 10443 filtinf 10973 xrmaxltsup 11684 fprodge0 12063 fprodge1 12065 xblss2ps 14991