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Theorem pnfnlt 9785
Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
pnfnlt  |-  ( A  e.  RR*  ->  -. +oo  <  A )

Proof of Theorem pnfnlt
StepHypRef Expression
1 pnfnre 7997 . . . . . . 7  |- +oo  e/  RR
21neli 2444 . . . . . 6  |-  -. +oo  e.  RR
32intnanr 930 . . . . 5  |-  -.  ( +oo  e.  RR  /\  A  e.  RR )
43intnanr 930 . . . 4  |-  -.  (
( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )
5 pnfnemnf 8010 . . . . . 6  |- +oo  =/= -oo
65neii 2349 . . . . 5  |-  -. +oo  = -oo
76intnanr 930 . . . 4  |-  -.  ( +oo  = -oo  /\  A  = +oo )
84, 7pm3.2ni 813 . . 3  |-  -.  (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )
92intnanr 930 . . . 4  |-  -.  ( +oo  e.  RR  /\  A  = +oo )
106intnanr 930 . . . 4  |-  -.  ( +oo  = -oo  /\  A  e.  RR )
119, 10pm3.2ni 813 . . 3  |-  -.  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) )
128, 11pm3.2ni 813 . 2  |-  -.  (
( ( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  ( ( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) )
13 pnfxr 8008 . . 3  |- +oo  e.  RR*
14 ltxr 9773 . . 3  |-  ( ( +oo  e.  RR*  /\  A  e.  RR* )  ->  ( +oo  <  A  <->  ( (
( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) ) ) )
1513, 14mpan 424 . 2  |-  ( A  e.  RR*  ->  ( +oo  <  A  <->  ( ( ( ( +oo  e.  RR  /\  A  e.  RR )  /\ +oo  <RR  A )  \/  ( +oo  = -oo  /\  A  = +oo ) )  \/  (
( +oo  e.  RR  /\  A  = +oo )  \/  ( +oo  = -oo  /\  A  e.  RR ) ) ) ) )
1612, 15mtbiri 675 1  |-  ( A  e.  RR*  ->  -. +oo  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   class class class wbr 4003   RRcr 7809    <RR cltrr 7814   +oocpnf 7987   -oocmnf 7988   RR*cxr 7989    < clt 7990
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995
This theorem is referenced by:  pnfge  9787  xrltnsym  9791  xrlttr  9793  xrltso  9794  xltnegi  9833  xposdif  9880  qbtwnxr  10255  xrmaxiflemab  11250  xrmaxltsup  11261
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