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Theorem xrrebnd 11394
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 11358 . . 3  |-  ( A  e.  RR  -> -oo  <  A )
2 ltpnf 11356 . . 3  |-  ( A  e.  RR  ->  A  < +oo )
31, 2jca 532 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  /\  A  < +oo ) )
4 nltpnft 11392 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
5 ngtmnft 11393 . . . . . 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 11350 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
13 3orass 976 . . . . . 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 972    = wceq 1395    e. wcel 1819   class class class wbr 4456   RRcr 9508   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651
This theorem is referenced by:  xrre  11395  xrre2  11396  xrre3  11397  supxrre1  11547  elioc2  11612  elico2  11613  elicc2  11614  xblpnfps  21023  xblpnf  21024  isnghm3  21357  ovoliun  22041  ovolicopnf  22060  voliunlem3  22087  volsup  22091  itg2seq  22274  nmblore  25827  nmopre  26915  limsupre  31808
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