Theorem ltpnf 11327

Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ltpnf	|- ( A e. RR -> A < +oo )

Proof of Theorem ltpnf

Step	Hyp	Ref	Expression
1		eqid 2467	. . . 4 |- +oo = +oo
2		orc 385	. . . 4 |- ( ( A e. RR /\ +oo = +oo ) -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
3	1, 2	mpan2 671	. . 3 |- ( A e. RR -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
4	3	olcd 393	. 2 |- ( A e. RR -> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) )
5		rexr 9635	. . 3 |- ( A e. RR -> A e. RR* )
6		pnfxr 11317	. . 3 |- +oo e. RR*
7		ltxr 11320	. . 3 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A < +oo <-> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) ) )
8	5, 6, 7	sylancl 662	. 2 |- ( A e. RR -> ( A < +oo <-> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) ) )
9	4, 8	mpbird 232	1 |- ( A e. RR -> A < +oo )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   class class class wbr 4447   RRcr 9487   <RR cltrr 9492   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623   < clt 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-pnf 9626  df-xr 9628  df-ltxr 9629

This theorem is referenced by:  0ltpnf  11328  xrlttri  11341  xrlttr  11342  xrrebnd  11365  xrre  11366  qbtwnxr  11395  xltnegi  11411  xrinfmsslem  11495  xrub  11499  supxrunb1  11507  supxrunb2  11508  elioc2  11583  elicc2  11585  ioomax  11595  ioopos  11597  elioopnf  11614  elicopnf  11616  difreicc  11648  hashbnd  12373  hashnnn0genn0  12378  hashv01gt1  12380  limsupgre  13260  pcadd  14260  ramubcl  14388  rge0srg  18252  mnfnei  19485  xblss2ps  20636  icopnfcld  21007  iocmnfcld  21008  blcvx  21035  xrtgioo  21043  reconnlem1  21063  xrge0tsms  21071  iccpnfhmeo  21177  ioombl1lem4  21703  icombl1  21705  uniioombllem1  21722  mbfmax  21788  ismbf3d  21793  mbflimsup  21805  itg2seq  21881  lhop2  22148  dvfsumlem2  22160  logccv  22769  xrlimcnp  23023  pntleme  23518  umgrafi  23995  frgrawopreglem2  24719  isblo3i  25389  htthlem  25507  xlt2addrd  27243  fsumrp0cl  27344  pnfinf  27386  xrge0tsmsd  27435  xrge0slmod  27494  xrge0iifcnv  27548  xrge0iifiso  27550  xrge0iifhom  27552  lmxrge0  27567  esumcst  27708  voliune  27838  volfiniune  27839  sxbrsigalem0  27879  orvcgteel  28043  dstfrvclim1  28053  itg2addnclem2  29642  asindmre  29677  dvasin  29678  dvacos  29679  rfcnpre3  30986  ltpnfd  31057  limcicciooub  31179  limsupre  31183  icccncfext  31226  cncfiooicclem1  31232  fourierdlem33  31440  fourierdlem75  31482  fourierdlem107  31514  fourierdlem109  31516  fourierdlem111  31518  fourierdlem113  31520  fouriersw  31532