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Theorem xrre2 8964
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 8943 . . . . . . 7  |-  ( A  e.  RR*  -> -oo  <_  A )
21adantr 270 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> -oo  <_  A )
3 mnfxr 7237 . . . . . . 7  |- -oo  e.  RR*
4 xrlelttr 8952 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( -oo  <_  A  /\  A  <  B )  -> -oo  <  B ) )
53, 4mp3an1 1256 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( -oo  <_  A  /\  A  <  B )  -> -oo  <  B ) )
62, 5mpand 420 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  -> -oo  <  B ) )
763adant3 959 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  -> -oo  <  B ) )
8 pnfge 8940 . . . . . . 7  |-  ( C  e.  RR*  ->  C  <_ +oo )
98adantl 271 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  C  <_ +oo )
10 pnfxr 7233 . . . . . . 7  |- +oo  e.  RR*
11 xrltletr 8953 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( B  <  C  /\  C  <_ +oo )  ->  B  < +oo ) )
1210, 11mp3an3 1258 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( B  <  C  /\  C  <_ +oo )  ->  B  < +oo )
)
139, 12mpan2d 419 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <  C  ->  B  < +oo ) )
14133adant1 957 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  <  C  ->  B  < +oo ) )
157, 14anim12d 328 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  ( -oo  <  B  /\  B  < +oo ) ) )
16 xrrebnd 8962 . . . 4  |-  ( B  e.  RR*  ->  ( B  e.  RR  <->  ( -oo  <  B  /\  B  < +oo ) ) )
17163ad2ant2 961 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  RR  <->  ( -oo  <  B  /\  B  < +oo ) ) )
1815, 17sylibrd 167 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  B  e.  RR ) )
1918imp 122 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3793   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214    < clt 7215    <_ cle 7216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-po 4059  df-iso 4060  df-xp 4377  df-cnv 4379  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221
This theorem is referenced by:  elioore  9011
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