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Theorem xrrebnd 9570
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 9531 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 id 19 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR )
3 mnflt 9537 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
4 ltpnf 9535 . . . . 5  |-  ( A  e.  RR  ->  A  < +oo )
53, 4jca 304 . . . 4  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
62, 52thd 174 . . 3  |-  ( A  e.  RR  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
7 renepnf 7781 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
87necon2bi 2340 . . . 4  |-  ( A  = +oo  ->  -.  A  e.  RR )
9 pnfxr 7786 . . . . . . 7  |- +oo  e.  RR*
10 xrltnr 9534 . . . . . . 7  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
119, 10ax-mp 5 . . . . . 6  |-  -. +oo  < +oo
12 breq1 3902 . . . . . 6  |-  ( A  = +oo  ->  ( A  < +oo  <-> +oo  < +oo )
)
1311, 12mtbiri 649 . . . . 5  |-  ( A  = +oo  ->  -.  A  < +oo )
1413intnand 901 . . . 4  |-  ( A  = +oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
158, 142falsed 676 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
16 renemnf 7782 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
1716necon2bi 2340 . . . 4  |-  ( A  = -oo  ->  -.  A  e.  RR )
18 mnfxr 7790 . . . . . . 7  |- -oo  e.  RR*
19 xrltnr 9534 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2018, 19ax-mp 5 . . . . . 6  |-  -. -oo  < -oo
21 breq2 3903 . . . . . 6  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2220, 21mtbiri 649 . . . . 5  |-  ( A  = -oo  ->  -. -oo 
<  A )
2322intnanrd 902 . . . 4  |-  ( A  = -oo  ->  -.  ( -oo  <  A  /\  A  < +oo ) )
2417, 232falsed 676 . . 3  |-  ( A  = -oo  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
256, 15, 243jaoi 1266 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
261, 25sylbi 120 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 946    = wceq 1316    e. wcel 1465   class class class wbr 3899   RRcr 7587   +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-in1 588  ax-in2 589  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-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  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-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  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-mnf 7771  df-xr 7772  df-ltxr 7773
This theorem is referenced by:  xrre  9571  xrre2  9572  xrre3  9573  elioc2  9687  elico2  9688  elicc2  9689  xblpnfps  12494  xblpnf  12495
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