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Theorem ltpnf 9350
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 2095 . . . 4  |- +oo  = +oo
2 orc 671 . . . 4  |-  ( ( A  e.  RR  /\ +oo  = +oo )  -> 
( ( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
31, 2mpan2 417 . . 3  |-  ( A  e.  RR  ->  (
( A  e.  RR  /\ +oo  = +oo )  \/  ( A  = -oo  /\ +oo  e.  RR ) ) )
43olcd 691 . 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 7630 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
6 pnfxr 7637 . . 3  |- +oo  e.  RR*
7 ltxr 9345 . . 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 405 . 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 667    = wceq 1296    e. wcel 1445   class class class wbr 3867   RRcr 7446    <RR cltrr 7451   +oocpnf 7616   -oocmnf 7617   RR*cxr 7618    < clt 7619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-pnf 7621  df-xr 7623  df-ltxr 7624
This theorem is referenced by:  0ltpnf  9351  xrlttr  9364  xrltso  9365  xrlttri3  9366  nltpnft  9380  npnflt  9381  xrrebnd  9385  xrre  9386  xltnegi  9401  xltadd1  9442  xposdif  9448  elioc2  9502  elicc2  9504  ioomax  9514  ioopos  9516  elioopnf  9533  elicopnf  9535  qbtwnxr  9818  filtinf  10315  xrmaxltsup  10801  xblss2ps  12190
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