ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltpnf Unicode version

Theorem ltpnf 9793
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
StepHypRef Expression
1 eqid 2187 . . . 4  |- +oo  = +oo
2 orc 713 . . . 4  |-  ( ( A  e.  RR  /\ +oo  = +oo )  -> 
( ( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
31, 2mpan2 425 . . 3  |-  ( A  e.  RR  ->  (
( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
43olcd 735 . 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 8016 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
6 pnfxr 8023 . . 3  |- +oo  e.  RR*
7 ltxr 9788 . . 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 ) ) ) ) )
85, 6, 7sylancl 413 . 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 ) ) ) ) )
94, 8mpbird 167 1  |-  ( A  e.  RR  ->  A  < +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1363    e. wcel 2158   class class class wbr 4015   RRcr 7823    <RR cltrr 7828   +oocpnf 8002   -oocmnf 8003   RR*cxr 8004    < clt 8005
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-pnf 8007  df-xr 8009  df-ltxr 8010
This theorem is referenced by:  ltpnfd  9794  0ltpnf  9795  xrlttr  9808  xrltso  9809  xrlttri3  9810  nltpnft  9827  npnflt  9828  xrrebnd  9832  xrre  9833  xltnegi  9848  xltadd1  9889  xposdif  9895  elioc2  9949  elicc2  9951  ioomax  9961  ioopos  9963  elioopnf  9980  elicopnf  9982  qbtwnxr  10271  dfrp2  10277  filtinf  10784  xrmaxltsup  11279  fprodge0  11658  fprodge1  11660  xblss2ps  14175
  Copyright terms: Public domain W3C validator