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Theorem xrre 9882
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 708 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  -oo  <  A )
2 ltpnf 9847 . . . . . 6  |-  ( B  e.  RR  ->  B  <  +oo )
32adantl 446 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  <  +oo )
4 rexr 8303 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 9839 . . . . . . 7  |-  +oo  e.  RR*
6 xrlelttr 9871 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo ) )
75, 6mp3an3 1224 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo )
)
84, 7sylan2 454 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo )
)
93, 8mpan2d 648 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  <  +oo ) )
109imp 415 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  <  +oo )
1110adantrl 687 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  A  <  +oo )
12 xrrebnd 9881 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
1312ad2antrr 697 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
141, 11, 13mpbir2and 850 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    /\ wa 356    e. wcel 1522   class class class wbr 3592   RRcr 8163    <_ cle 8290    +oocpnf 8291    -oocmnf 8292   RR*cxr 8293    < clt 8294
This theorem is referenced by:  xrrege0  9886  supxrre  10029  infmxrre  10037  caucvgrlem  11352  pcgcd1  12050  tgioo  17007  ovolunlem1a  17552  ovoliunlem1  17558  ioombl1lem2  17613  itg2monolem2  17803  dvferm1lem  18002  radcnvle  18401  psercnlem1  18406  nmobndi  19905  ubthlem3  20003  nmophmi  21163  bdophsi  21228  bdopcoi  21230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-sep 3707  ax-nul 3715  ax-pow 3751  ax-pr 3775  ax-un 4067  ax-cnex 8219  ax-resscn 8220  ax-pre-lttri 8237  ax-pre-lttrn 8238
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1447  df-sb 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-nel 2186  df-ral 2279  df-rex 2280  df-rab 2282  df-v 2478  df-sbc 2652  df-csb 2734  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-op 3265  df-uni 3431  df-br 3593  df-opab 3647  df-mpt 3648  df-id 3866  df-po 3871  df-so 3872  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-fun 4284  df-fn 4285  df-f 4286  df-f1 4287  df-fo 4288  df-f1o 4289  df-fv 4290  df-er 6121  df-en 6308  df-dom 6309  df-sdom 6310  df-pnf 8295  df-mnf 8296  df-xr 8297  df-ltxr 8298  df-le 8299
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