ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9782

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 8005	. . 3
6		pnfxr 8012	. . 3
7		ltxr 9777	. . 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 4005

cr 7812

cltrr 7817

cpnf 7991

cmnf 7992

cxr 7993

clt 7994

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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904

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 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-pnf 7996 df-xr 7998 df-ltxr 7999

This theorem is referenced by: ltpnfd 9783 0ltpnf 9784 xrlttr 9797 xrltso 9798 xrlttri3 9799 nltpnft 9816 npnflt 9817 xrrebnd 9821 xrre 9822 xltnegi 9837 xltadd1 9878 xposdif 9884 elioc2 9938 elicc2 9940 ioomax 9950 ioopos 9952 elioopnf 9969 elicopnf 9971 qbtwnxr 10260 dfrp2 10266 filtinf 10773 xrmaxltsup 11268 fprodge0 11647 fprodge1 11649 xblss2ps 13989