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Theorem xrrebnd 8962
Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
Assertion
Ref Expression
xrrebnd  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )

Proof of Theorem xrrebnd
StepHypRef Expression
1 elxr 8928 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 id 19 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR )
3 mnflt 8934 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
4 ltpnf 8932 . . . . 5  |-  ( A  e.  RR  ->  A  < +oo )
53, 4jca 300 . . . 4  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
62, 52thd 173 . . 3  |-  ( A  e.  RR  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
7 renepnf 7228 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
87necon2bi 2301 . . . 4  |-  ( A  = +oo  ->  -.  A  e.  RR )
9 pnfxr 7233 . . . . . . 7  |- +oo  e.  RR*
10 xrltnr 8931 . . . . . . 7  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
119, 10ax-mp 7 . . . . . 6  |-  -. +oo  < +oo
12 breq1 3796 . . . . . 6  |-  ( A  = +oo  ->  ( A  < +oo  <-> +oo  < +oo )
)
1311, 12mtbiri 633 . . . . 5  |-  ( A  = +oo  ->  -.  A  < +oo )
1413intnand 874 . . . 4  |-  ( A  = +oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
158, 142falsed 651 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
16 renemnf 7229 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
1716necon2bi 2301 . . . 4  |-  ( A  = -oo  ->  -.  A  e.  RR )
18 mnfxr 7237 . . . . . . 7  |- -oo  e.  RR*
19 xrltnr 8931 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2018, 19ax-mp 7 . . . . . 6  |-  -. -oo  < -oo
21 breq2 3797 . . . . . 6  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2220, 21mtbiri 633 . . . . 5  |-  ( A  = -oo  ->  -. -oo 
<  A )
2322intnanrd 875 . . . 4  |-  ( A  = -oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
2417, 232falsed 651 . . 3  |-  ( A  = -oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
256, 15, 243jaoi 1235 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
261, 25sylbi 119 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 919    = wceq 1285    e. wcel 1434   class class class wbr 3793   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214    < clt 7215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-pre-ltirr 7150
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220
This theorem is referenced by:  xrre  8963  xrre2  8964  xrre3  8965  elioc2  9035  elico2  9036  elicc2  9037
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