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Theorem xrre3 9852
Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
xrre3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  e.  RR )

Proof of Theorem xrre3
StepHypRef Expression
1 mnflt 9813 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
21adantl 277 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  <  B )
3 mnfxr 8044 . . . . . . 7  |- -oo  e.  RR*
43a1i 9 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  e.  RR* )
5 rexr 8033 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  RR* )
65adantl 277 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  e.  RR* )
7 simpl 109 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  A  e.  RR* )
8 xrltletr 9837 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  A  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
94, 6, 7, 8syl3anc 1249 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
102, 9mpand 429 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( B  <_  A  -> -oo  <  A ) )
1110imp 124 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  B  <_  A
)  -> -oo  <  A
)
1211adantrr 479 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  -> -oo  <  A )
13 simprr 531 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  < +oo )
14 xrrebnd 9849 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1514ad2antrr 488 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1612, 13, 15mpbir2and 946 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160   class class class wbr 4018   RRcr 7840   +oocpnf 8019   -oocmnf 8020   RR*cxr 8021    < clt 8022    <_ cle 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-po 4314  df-iso 4315  df-xp 4650  df-cnv 4652  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028
This theorem is referenced by:  elicore  10297
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