Theorem ltpnf 11302

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 2402	. . . 4 |- +oo = +oo
2		orc 383	. . . 4 |- ( ( A e. RR /\ +oo = +oo ) -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
3	1, 2	mpan2 669	. . 3 |- ( A e. RR -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
4	3	olcd 391	. 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 9589	. . 3 |- ( A e. RR -> A e. RR* )
6		pnfxr 11292	. . 3 |- +oo e. RR*
7		ltxr 11295	. . 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 660	. 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 366   /\ wa 367   = wceq 1405   e. wcel 1842   class class class wbr 4394   RRcr 9441   <RR cltrr 9446   +oocpnf 9575   -oocmnf 9576   RR*cxr 9577   < clt 9578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-pnf 9580  df-xr 9582  df-ltxr 9583

This theorem is referenced by:  0ltpnf  11303  xrlttri  11316  xrlttr  11317  xrrebnd  11340  xrre  11341  qbtwnxr  11370  xltnegi  11386  xrinfmsslem  11470  xrub  11474  supxrunb1  11482  supxrunb2  11483  elioc2  11558  elicc2  11560  ioomax  11570  ioopos  11572  elioopnf  11589  elicopnf  11591  difreicc  11623  hashbnd  12365  hashnnn0genn0  12370  hashv01gt1  12372  limsupgre  13360  pcadd  14509  ramubcl  14637  rge0srg  18699  mnfnei  19907  xblss2ps  21088  icopnfcld  21459  iocmnfcld  21460  blcvx  21487  xrtgioo  21495  reconnlem1  21515  xrge0tsms  21523  iccpnfhmeo  21629  ioombl1lem4  22155  icombl1  22157  uniioombllem1  22174  mbfmax  22240  ismbf3d  22245  mbflimsup  22257  itg2seq  22333  lhop2  22600  dvfsumlem2  22612  logccv  23230  xrlimcnp  23516  pntleme  24066  umgrafi  24620  frgrawopreglem2  25343  isblo3i  26010  htthlem  26128  xlt2addrd  27900  dfrp2  27909  fsumrp0cl  28017  pnfinf  28059  xrge0tsmsd  28108  xrge0slmod  28167  xrge0iifcnv  28248  xrge0iifiso  28250  xrge0iifhom  28252  lmxrge0  28267  esumcst  28390  esumcvgre  28418  voliune  28558  volfiniune  28559  sxbrsigalem0  28599  orvcgteel  28792  dstfrvclim1  28802  topnfbey  28945  itg2addnclem2  31421  asindmre  31454  dvasin  31455  dvacos  31456  rfcnpre3  36769  ltpnfd  36833  fprodge0  36947  fprodge1  36948  limsupre  36997  icccncfext  37040  fourierdlem111  37350  fourierdlem113  37352  fouriersw  37364  iccpartiltu  37669