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Theorem ltpnf 9522
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 2117 . . . 4  |- +oo  = +oo
2 orc 686 . . . 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 708 . 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 7779 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
6 pnfxr 7786 . . 3  |- +oo  e.  RR*
7 ltxr 9517 . . 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 682    = wceq 1316    e. wcel 1465   class class class wbr 3899   RRcr 7587    <RR cltrr 7592   +oocpnf 7765   -oocmnf 7766   RR*cxr 7767    < clt 7768
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-pnf 7770  df-xr 7772  df-ltxr 7773
This theorem is referenced by:  0ltpnf  9523  xrlttr  9536  xrltso  9537  xrlttri3  9538  nltpnft  9552  npnflt  9553  xrrebnd  9557  xrre  9558  xltnegi  9573  xltadd1  9614  xposdif  9620  elioc2  9674  elicc2  9676  ioomax  9686  ioopos  9688  elioopnf  9705  elicopnf  9707  qbtwnxr  9990  filtinf  10493  xrmaxltsup  10982  xblss2ps  12484
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