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Theorem xrrebnd 11155
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 11119 . . 3  |-  ( A  e.  RR  -> -oo  <  A )
2 ltpnf 11117 . . 3  |-  ( A  e.  RR  ->  A  < +oo )
31, 2jca 532 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
4 nltpnft 11153 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
5 ngtmnft 11154 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
64, 5orbi12d 709 . . . . 5  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  <->  ( -.  A  < +oo  \/  -. -oo  <  A
) ) )
7 ianor 488 . . . . . 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 11111 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
13 3orass 968 . . . . . 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 964    = wceq 1369    e. wcel 1756   class class class wbr 4307   RRcr 9296   +oocpnf 9430   -oocmnf 9431   RR*cxr 9432    < clt 9433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-pre-lttri 9371  ax-pre-lttrn 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439
This theorem is referenced by:  xrre  11156  xrre2  11157  xrre3  11158  supxrre1  11308  elioc2  11373  elico2  11374  elicc2  11375  xblpnfps  19985  xblpnf  19986  isnghm3  20319  ovoliun  21003  ovolicopnf  21022  voliunlem3  21048  volsup  21052  itg2seq  21235  nmblore  24201  nmopre  25289
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