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Theorem xrre 9554
Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
Assertion
Ref Expression
xrre  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )

Proof of Theorem xrre
StepHypRef Expression
1 simprl 503 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  -> -oo  <  A )
2 ltpnf 9518 . . . . . 6  |-  ( B  e.  RR  ->  B  < +oo )
32adantl 273 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  < +oo )
4 rexr 7775 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 7782 . . . . . . 7  |- +oo  e.  RR*
6 xrlelttr 9540 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  < +oo )  ->  A  < +oo ) )
75, 6mp3an3 1287 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
84, 7sylan2 282 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
93, 8mpan2d 422 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  < +oo ) )
109imp 123 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  < +oo )
1110adantrl 467 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  < +oo )
12 xrrebnd 9553 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1312ad2antrr 477 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
141, 11, 13mpbir2and 911 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1463   class class class wbr 3897   RRcr 7583   +oocpnf 7761   -oocmnf 7762   RR*cxr 7763    < clt 7764    <_ cle 7765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-po 4186  df-iso 4187  df-xp 4513  df-cnv 4515  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770
This theorem is referenced by:  xrrege0  9559  tgioo  12621
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