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Theorem xrrebnd 11136
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 mnflt 11100 . . 3  |-  ( A  e.  RR  -> -oo  <  A )
2 ltpnf 11098 . . 3  |-  ( A  e.  RR  ->  A  < +oo )
31, 2jca 529 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
4 nltpnft 11134 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
5 ngtmnft 11135 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
64, 5orbi12d 704 . . . . 5  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  <->  ( -.  A  < +oo  \/  -. -oo  <  A
) ) )
7 ianor 485 . . . . . 6  |-  ( -.  ( -oo  <  A  /\  A  < +oo )  <->  ( -. -oo  <  A  \/  -.  A  < +oo ) )
8 orcom 387 . . . . . 6  |-  ( ( -. -oo  <  A  \/  -.  A  < +oo ) 
<->  ( -.  A  < +oo  \/  -. -oo  <  A ) )
97, 8bitr2i 250 . . . . 5  |-  ( ( -.  A  < +oo  \/  -. -oo  <  A
)  <->  -.  ( -oo  <  A  /\  A  < +oo ) )
106, 9syl6bb 261 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  <->  -.  ( -oo  <  A  /\  A  < +oo )
) )
1110con2bid 329 . . 3  |-  ( A  e.  RR*  ->  ( ( -oo  <  A  /\  A  < +oo )  <->  -.  ( A  = +oo  \/  A  = -oo ) ) )
12 elxr 11092 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
13 3orass 963 . . . . . 6  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) ) )
14 orcom 387 . . . . . 6  |-  ( ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) )  <->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1513, 14bitri 249 . . . . 5  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1612, 15sylbb 197 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1716ord 377 . . 3  |-  ( A  e.  RR*  ->  ( -.  ( A  = +oo  \/  A  = -oo )  ->  A  e.  RR ) )
1811, 17sylbid 215 . 2  |-  ( A  e.  RR*  ->  ( ( -oo  <  A  /\  A  < +oo )  ->  A  e.  RR ) )
193, 18impbid2 204 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 959    = wceq 1364    e. wcel 1761   class class class wbr 4289   RRcr 9277   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413    < clt 9414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420
This theorem is referenced by:  xrre  11137  xrre2  11138  xrre3  11139  supxrre1  11289  elioc2  11354  elico2  11355  elicc2  11356  xblpnfps  19929  xblpnf  19930  isnghm3  20263  ovoliun  20947  ovolicopnf  20966  voliunlem3  20992  volsup  20996  itg2seq  21179  nmblore  24121  nmopre  25209
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