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Theorem xrrebnd 10689
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 10655 . . 3  |-  ( A  e.  RR  ->  -oo  <  A )
2 ltpnf 10654 . . 3  |-  ( A  e.  RR  ->  A  <  +oo )
31, 2jca 519 . 2  |-  ( A  e.  RR  ->  (  -oo  <  A  /\  A  <  +oo ) )
4 nltpnft 10687 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  +oo  <->  -.  A  <  +oo ) )
5 ngtmnft 10688 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  -oo  <->  -.  -oo  <  A ) )
64, 5orbi12d 691 . . . . 5  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  <->  ( -.  A  <  +oo  \/  -.  -oo 
<  A ) ) )
7 ianor 475 . . . . . 6  |-  ( -.  (  -oo  <  A  /\  A  <  +oo )  <->  ( -.  -oo  <  A  \/  -.  A  <  +oo )
)
8 orcom 377 . . . . . 6  |-  ( ( -.  -oo  <  A  \/  -.  A  <  +oo )  <->  ( -.  A  <  +oo  \/ 
-.  -oo  <  A ) )
97, 8bitr2i 242 . . . . 5  |-  ( ( -.  A  <  +oo  \/ 
-.  -oo  <  A )  <->  -.  (  -oo  <  A  /\  A  <  +oo )
)
106, 9syl6bb 253 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  <->  -.  (  -oo  <  A  /\  A  <  +oo ) ) )
1110con2bid 320 . . 3  |-  ( A  e.  RR*  ->  ( ( 
-oo  <  A  /\  A  <  +oo )  <->  -.  ( A  =  +oo  \/  A  =  -oo ) ) )
12 elxr 10649 . . . . . 6  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
1312biimpi 187 . . . . 5  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  =  +oo  \/  A  = 
-oo ) )
14 3orass 939 . . . . . 6  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo ) ) )
15 orcom 377 . . . . . 6  |-  ( ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo )
)  <->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1614, 15bitri 241 . . . . 5  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1713, 16sylib 189 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1817ord 367 . . 3  |-  ( A  e.  RR*  ->  ( -.  ( A  =  +oo  \/  A  =  -oo )  ->  A  e.  RR ) )
1911, 18sylbid 207 . 2  |-  ( A  e.  RR*  ->  ( ( 
-oo  <  A  /\  A  <  +oo )  ->  A  e.  RR ) )
203, 19impbid2 196 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   class class class wbr 4154   RRcr 8923    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053    < clt 9054
This theorem is referenced by:  xrre  10690  xrre2  10691  xrre3  10692  supxrre1  10842  elioc2  10906  elico2  10907  elicc2  10908  xblpnf  18328  isnghm3  18631  ovoliun  19269  ovolicopnf  19288  voliunlem3  19314  volsup  19318  itg2seq  19502  nmblore  22136  nmopre  23222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-pre-lttri 8998  ax-pre-lttrn 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060
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