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Theorem xrrebnd 11471
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 11433 . . 3  |-  ( A  e.  RR  -> -oo  <  A )
2 ltpnf 11430 . . 3  |-  ( A  e.  RR  ->  A  < +oo )
31, 2jca 534 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
4 nltpnft 11469 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
5 ngtmnft 11470 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
64, 5orbi12d 714 . . . . 5  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  <->  ( -.  A  < +oo  \/  -. -oo  <  A
) ) )
7 ianor 490 . . . . . 6  |-  ( -.  ( -oo  <  A  /\  A  < +oo )  <->  ( -. -oo  <  A  \/  -.  A  < +oo ) )
8 orcom 388 . . . . . 6  |-  ( ( -. -oo  <  A  \/  -.  A  < +oo ) 
<->  ( -.  A  < +oo  \/  -. -oo  <  A ) )
97, 8bitr2i 253 . . . . 5  |-  ( ( -.  A  < +oo  \/  -. -oo  <  A
)  <->  -.  ( -oo  <  A  /\  A  < +oo ) )
106, 9syl6bb 264 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  <->  -.  ( -oo  <  A  /\  A  < +oo )
) )
1110con2bid 330 . . 3  |-  ( A  e.  RR*  ->  ( ( -oo  <  A  /\  A  < +oo )  <->  -.  ( A  = +oo  \/  A  = -oo ) ) )
12 elxr 11424 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
13 3orass 985 . . . . . 6  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) ) )
14 orcom 388 . . . . . 6  |-  ( ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) )  <->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1513, 14bitri 252 . . . . 5  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1612, 15sylbb 200 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  = +oo  \/  A  = -oo )  \/  A  e.  RR ) )
1716ord 378 . . 3  |-  ( A  e.  RR*  ->  ( -.  ( A  = +oo  \/  A  = -oo )  ->  A  e.  RR ) )
1811, 17sylbid 218 . 2  |-  ( A  e.  RR*  ->  ( ( -oo  <  A  /\  A  < +oo )  ->  A  e.  RR ) )
193, 18impbid2 207 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872   class class class wbr 4423   RRcr 9546   +oocpnf 9680   -oocmnf 9681   RR*cxr 9682    < clt 9683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-pre-lttri 9621  ax-pre-lttrn 9622
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689
This theorem is referenced by:  xrre  11472  xrre2  11473  xrre3  11474  supxrre1  11624  elioc2  11705  elico2  11706  elicc2  11707  xblpnfps  21409  xblpnf  21410  isnghm3  21729  isnghm3OLD  21747  ovoliun  22457  ovolicopnf  22477  voliunlem3  22504  volsup  22508  itg2seq  22699  nmblore  26426  nmopre  27522  supxrgere  37511  supxrgelem  37515  supxrge  37516  suplesup  37517  infrpge  37529  limsupre  37662  limsupreOLD  37663
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