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Theorem xrre3 9216
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 9185 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
21adantl 271 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  <  B )
3 mnfxr 7488 . . . . . . 7  |- -oo  e.  RR*
43a1i 9 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  e.  RR* )
5 rexr 7477 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  RR* )
65adantl 271 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  e.  RR* )
7 simpl 107 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  A  e.  RR* )
8 xrltletr 9204 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  A  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
94, 6, 7, 8syl3anc 1172 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
102, 9mpand 420 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( B  <_  A  -> -oo  <  A ) )
1110imp 122 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  B  <_  A
)  -> -oo  <  A
)
1211adantrr 463 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  -> -oo  <  A )
13 simprr 499 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  < +oo )
14 xrrebnd 9213 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1514ad2antrr 472 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1612, 13, 15mpbir2and 888 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 102    <-> wb 103    e. wcel 1436   class class class wbr 3820   RRcr 7293   +oocpnf 7463   -oocmnf 7464   RR*cxr 7465    < clt 7466    <_ cle 7467
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-po 4097  df-iso 4098  df-xp 4417  df-cnv 4419  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472
This theorem is referenced by: (None)
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