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Theorem xrrebnd 10748
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 10714 . . 3  |-  ( A  e.  RR  ->  -oo  <  A )
2 ltpnf 10713 . . 3  |-  ( A  e.  RR  ->  A  <  +oo )
31, 2jca 519 . 2  |-  ( A  e.  RR  ->  (  -oo  <  A  /\  A  <  +oo ) )
4 nltpnft 10746 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  +oo  <->  -.  A  <  +oo ) )
5 ngtmnft 10747 . . . . . 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 10708 . . . . . 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 1652    e. wcel 1725   class class class wbr 4204   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112
This theorem is referenced by:  xrre  10749  xrre2  10750  xrre3  10751  supxrre1  10901  elioc2  10965  elico2  10966  elicc2  10967  xblpnfps  18417  xblpnf  18418  isnghm3  18751  ovoliun  19393  ovolicopnf  19412  voliunlem3  19438  volsup  19442  itg2seq  19626  nmblore  22279  nmopre  23365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118
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