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Theorem ltpnf 9574
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 2139 . . . 4  |- +oo  = +oo
2 orc 701 . . . 4  |-  ( ( A  e.  RR  /\ +oo  = +oo )  -> 
( ( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
31, 2mpan2 421 . . 3  |-  ( A  e.  RR  ->  (
( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
43olcd 723 . 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 7818 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
6 pnfxr 7825 . . 3  |- +oo  e.  RR*
7 ltxr 9569 . . 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 409 . 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 166 1  |-  ( A  e.  RR  ->  A  < +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   class class class wbr 3929   RRcr 7626    <RR cltrr 7631   +oocpnf 7804   -oocmnf 7805   RR*cxr 7806    < clt 7807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-cnex 7718
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7809  df-xr 7811  df-ltxr 7812
This theorem is referenced by:  0ltpnf  9575  xrlttr  9588  xrltso  9589  xrlttri3  9590  nltpnft  9604  npnflt  9605  xrrebnd  9609  xrre  9610  xltnegi  9625  xltadd1  9666  xposdif  9672  elioc2  9726  elicc2  9728  ioomax  9738  ioopos  9740  elioopnf  9757  elicopnf  9759  qbtwnxr  10042  filtinf  10545  xrmaxltsup  11034  xblss2ps  12582
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