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Theorem xrre2 11142
Description: An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
Assertion
Ref Expression
xrre2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR )

Proof of Theorem xrre2
StepHypRef Expression
1 mnfle 11113 . . . . . . 7  |-  ( A  e.  RR*  -> -oo  <_  A )
21adantr 465 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> -oo  <_  A )
3 mnfxr 11094 . . . . . . 7  |- -oo  e.  RR*
4 xrlelttr 11130 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( -oo  <_  A  /\  A  <  B )  -> -oo  <  B ) )
53, 4mp3an1 1301 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( -oo  <_  A  /\  A  <  B )  -> -oo  <  B ) )
62, 5mpand 675 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  -> -oo  <  B ) )
763adant3 1008 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  -> -oo  <  B ) )
8 pnfge 11110 . . . . . . 7  |-  ( C  e.  RR*  ->  C  <_ +oo )
98adantl 466 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  C  <_ +oo )
10 pnfxr 11092 . . . . . . 7  |- +oo  e.  RR*
11 xrltletr 11131 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( B  <  C  /\  C  <_ +oo )  ->  B  < +oo ) )
1210, 11mp3an3 1303 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( B  <  C  /\  C  <_ +oo )  ->  B  < +oo )
)
139, 12mpan2d 674 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <  C  ->  B  < +oo ) )
14133adant1 1006 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  <  C  ->  B  < +oo ) )
157, 14anim12d 563 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  ( -oo  <  B  /\  B  < +oo ) ) )
16 xrrebnd 11140 . . . 4  |-  ( B  e.  RR*  ->  ( B  e.  RR  <->  ( -oo  <  B  /\  B  < +oo ) ) )
17163ad2ant2 1010 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  RR  <->  ( -oo  <  B  /\  B  < +oo ) ) )
1815, 17sylibrd 234 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  B  e.  RR ) )
1918imp 429 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756   class class class wbr 4292   RRcr 9281   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424
This theorem is referenced by:  elioore  11330  xrsdsreclblem  17859  pnfnei  18824  mnfnei  18825  tgioo  20373  ovolunnul  20983  icombl  21045  ioombl  21046  ioorcl2  21052  volivth  21087  dvferm2lem  21458  itg2gt0cn  28447
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