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Theorem xrrebnd 9279
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 9245 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 id 19 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR )
3 mnflt 9251 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
4 ltpnf 9249 . . . . 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 7533 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
87necon2bi 2310 . . . 4  |-  ( A  = +oo  ->  -.  A  e.  RR )
9 pnfxr 7538 . . . . . . 7  |- +oo  e.  RR*
10 xrltnr 9248 . . . . . . 7  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
119, 10ax-mp 7 . . . . . 6  |-  -. +oo  < +oo
12 breq1 3848 . . . . . 6  |-  ( A  = +oo  ->  ( A  < +oo  <-> +oo  < +oo )
)
1311, 12mtbiri 635 . . . . 5  |-  ( A  = +oo  ->  -.  A  < +oo )
1413intnand 878 . . . 4  |-  ( A  = +oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
158, 142falsed 653 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
16 renemnf 7534 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
1716necon2bi 2310 . . . 4  |-  ( A  = -oo  ->  -.  A  e.  RR )
18 mnfxr 7542 . . . . . . 7  |- -oo  e.  RR*
19 xrltnr 9248 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2018, 19ax-mp 7 . . . . . 6  |-  -. -oo  < -oo
21 breq2 3849 . . . . . 6  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2220, 21mtbiri 635 . . . . 5  |-  ( A  = -oo  ->  -. -oo 
<  A )
2322intnanrd 879 . . . 4  |-  ( A  = -oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
2417, 232falsed 653 . . 3  |-  ( A  = -oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
256, 15, 243jaoi 1239 . 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 923    = wceq 1289    e. wcel 1438   class class class wbr 3845   RRcr 7347   +oocpnf 7517   -oocmnf 7518   RR*cxr 7519    < clt 7520
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-pre-ltirr 7455
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525
This theorem is referenced by:  xrre  9280  xrre2  9281  xrre3  9282  elioc2  9352  elico2  9353  elicc2  9354
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