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Theorem xrre2 11396
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 11367 . . . . . . 7  |-  ( A  e.  RR*  -> -oo  <_  A )
21adantr 465 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> -oo  <_  A )
3 mnfxr 11348 . . . . . . 7  |- -oo  e.  RR*
4 xrlelttr 11384 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( -oo  <_  A  /\  A  <  B )  -> -oo  <  B ) )
53, 4mp3an1 1311 . . . . . 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 1016 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  -> -oo  <  B ) )
8 pnfge 11364 . . . . . . 7  |-  ( C  e.  RR*  ->  C  <_ +oo )
98adantl 466 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  C  <_ +oo )
10 pnfxr 11346 . . . . . . 7  |- +oo  e.  RR*
11 xrltletr 11385 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( B  <  C  /\  C  <_ +oo )  ->  B  < +oo ) )
1210, 11mp3an3 1313 . . . . . 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 1014 . . . 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 11394 . . . 4  |-  ( B  e.  RR*  ->  ( B  e.  RR  <->  ( -oo  <  B  /\  B  < +oo ) ) )
17163ad2ant2 1018 . . 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 973    e. wcel 1819   class class class wbr 4456   RRcr 9508   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651
This theorem is referenced by:  elioore  11584  xrsdsreclblem  18591  pnfnei  19848  mnfnei  19849  tgioo  21427  ovolunnul  22037  icombl  22100  ioombl  22101  ioorcl2  22107  volivth  22142  dvferm2lem  22513  itg2gt0cn  30275
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