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Theorem xrrebnd 10499
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 10466 . . 3  |-  ( A  e.  RR  ->  -oo  <  A )
2 ltpnf 10465 . . 3  |-  ( A  e.  RR  ->  A  <  +oo )
31, 2jca 518 . 2  |-  ( A  e.  RR  ->  (  -oo  <  A  /\  A  <  +oo ) )
4 nltpnft 10497 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  +oo  <->  -.  A  <  +oo ) )
5 ngtmnft 10498 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  -oo  <->  -.  -oo  <  A ) )
64, 5orbi12d 690 . . . . 5  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  <->  ( -.  A  <  +oo  \/  -.  -oo 
<  A ) ) )
7 ianor 474 . . . . . 6  |-  ( -.  (  -oo  <  A  /\  A  <  +oo )  <->  ( -.  -oo  <  A  \/  -.  A  <  +oo )
)
8 orcom 376 . . . . . 6  |-  ( ( -.  -oo  <  A  \/  -.  A  <  +oo )  <->  ( -.  A  <  +oo  \/ 
-.  -oo  <  A ) )
97, 8bitr2i 241 . . . . 5  |-  ( ( -.  A  <  +oo  \/ 
-.  -oo  <  A )  <->  -.  (  -oo  <  A  /\  A  <  +oo )
)
106, 9syl6bb 252 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  <->  -.  (  -oo  <  A  /\  A  <  +oo ) ) )
1110con2bid 319 . . 3  |-  ( A  e.  RR*  ->  ( ( 
-oo  <  A  /\  A  <  +oo )  <->  -.  ( A  =  +oo  \/  A  =  -oo ) ) )
12 elxr 10460 . . . . . 6  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
1312biimpi 186 . . . . 5  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  =  +oo  \/  A  = 
-oo ) )
14 3orass 937 . . . . . 6  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo ) ) )
15 orcom 376 . . . . . 6  |-  ( ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo )
)  <->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1614, 15bitri 240 . . . . 5  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1713, 16sylib 188 . . . 4  |-  ( A  e.  RR*  ->  ( ( A  =  +oo  \/  A  =  -oo )  \/  A  e.  RR ) )
1817ord 366 . . 3  |-  ( A  e.  RR*  ->  ( -.  ( A  =  +oo  \/  A  =  -oo )  ->  A  e.  RR ) )
1911, 18sylbid 206 . 2  |-  ( A  e.  RR*  ->  ( ( 
-oo  <  A  /\  A  <  +oo )  ->  A  e.  RR ) )
203, 19impbid2 195 1  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1625    e. wcel 1686   class class class wbr 4025   RRcr 8738    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    < clt 8869
This theorem is referenced by:  xrre  10500  xrre2  10501  xrre3  10502  supxrre1  10651  elioc2  10715  elico2  10716  elicc2  10717  xblpnf  17955  isnghm3  18236  ovoliun  18866  ovolicopnf  18885  voliunlem3  18911  volsup  18915  itg2seq  19099  nmblore  21366  nmopre  22452  xrre3FL  23253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-pre-lttri 8813  ax-pre-lttrn 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875
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