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Theorem xrre 9251
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 498 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  -> -oo  <  A )
2 ltpnf 9220 . . . . . 6  |-  ( B  e.  RR  ->  B  < +oo )
32adantl 271 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  < +oo )
4 rexr 7512 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 7519 . . . . . . 7  |- +oo  e.  RR*
6 xrlelttr 9240 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  < +oo )  ->  A  < +oo ) )
75, 6mp3an3 1262 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
84, 7sylan2 280 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
93, 8mpan2d 419 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  < +oo ) )
109imp 122 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  < +oo )
1110adantrl 462 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  < +oo )
12 xrrebnd 9250 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1312ad2antrr 472 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
141, 11, 13mpbir2and 890 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 102    <-> wb 103    e. wcel 1438   class class class wbr 3837   RRcr 7328   +oocpnf 7498   -oocmnf 7499   RR*cxr 7500    < clt 7501    <_ cle 7502
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-po 4114  df-iso 4115  df-xp 4434  df-cnv 4436  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507
This theorem is referenced by:  xrrege0  9256
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