ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 8856

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 2081	. . . 4
2		orc 665	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan2 415	. . 3 $/\$ $\/$ $/\$
4	3	olcd 685	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		rexr 7164	. . 3
6		pnfxr 8846	. . 3
7		ltxr 8849	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancl 404	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 165	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661

wceq 1284

wcel 1433 class class class wbr 3785

cr 6980

cltrr 6985

cpnf 7150

cmnf 7151

cxr 7152

clt 7153

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-xr 7157 df-ltxr 7158

This theorem is referenced by: 0ltpnf 8857 xrlttr 8870 xrltso 8871 xrlttri3 8872 nltpnft 8884 xrrebnd 8886 xrre 8887 xltnegi 8902 elioc2 8959 elicc2 8961 ioomax 8971 ioopos 8973 elioopnf 8990 elicopnf 8992 qbtwnxr 9266