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Theorem xrre 9834
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 529 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  -> -oo  <  A )
2 ltpnf 9794 . . . . . 6  |-  ( B  e.  RR  ->  B  < +oo )
32adantl 277 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  < +oo )
4 rexr 8017 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 8024 . . . . . . 7  |- +oo  e.  RR*
6 xrlelttr 9820 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  < +oo )  ->  A  < +oo ) )
75, 6mp3an3 1336 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
84, 7sylan2 286 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
93, 8mpan2d 428 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  < +oo ) )
109imp 124 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  < +oo )
1110adantrl 478 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  < +oo )
12 xrrebnd 9833 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1312ad2antrr 488 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
141, 11, 13mpbir2and 945 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 104    <-> wb 105    e. wcel 2158   class class class wbr 4015   RRcr 7824   +oocpnf 8003   -oocmnf 8004   RR*cxr 8005    < clt 8006    <_ cle 8007
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-po 4308  df-iso 4309  df-xp 4644  df-cnv 4646  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012
This theorem is referenced by:  xrrege0  9839  pcgcd1  12341  tgioo  14342
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