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Theorem xrre 9886
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 9851 . . . . . 6  |-  ( B  e.  RR  ->  B  <  +oo )
32adantl 446 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  <  +oo )
4 rexr 8307 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 9843 . . . . . . 7  |-  +oo  e.  RR*
6 xrlelttr 9875 . . . . . . 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 9885 . . 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 1526   class class class wbr 3596   RRcr 8167    <_ cle 8294    +oocpnf 8295    -oocmnf 8296   RR*cxr 8297    < clt 8298
This theorem is referenced by:  xrrege0  9890  supxrre  10033  infmxrre  10041  caucvgrlem  11356  pcgcd1  12054  tgioo  17011  ovolunlem1a  17556  ovoliunlem1  17562  ioombl1lem2  17617  itg2monolem2  17807  dvferm1lem  18006  radcnvle  18405  psercnlem1  18410  nmobndi  19909  ubthlem3  20007  nmophmi  21167  bdophsi  21232  bdopcoi  21234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-sep 3711  ax-nul 3719  ax-pow 3755  ax-pr 3779  ax-un 4071  ax-cnex 8223  ax-resscn 8224  ax-pre-lttri 8241  ax-pre-lttrn 8242
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 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-nel 2190  df-ral 2283  df-rex 2284  df-rab 2286  df-v 2482  df-sbc 2656  df-csb 2738  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-nul 3078  df-if 3187  df-pw 3248  df-sn 3266  df-pr 3267  df-op 3269  df-uni 3435  df-br 3597  df-opab 3651  df-mpt 3652  df-id 3870  df-po 3875  df-so 3876  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fn 4289  df-f 4290  df-f1 4291  df-fo 4292  df-f1o 4293  df-fv 4294  df-er 6125  df-en 6312  df-dom 6313  df-sdom 6314  df-pnf 8299  df-mnf 8300  df-xr 8301  df-ltxr 8302  df-le 8303
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