ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrre Unicode version

Theorem xrre 9724
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 521 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  -> -oo  <  A )
2 ltpnf 9687 . . . . . 6  |-  ( B  e.  RR  ->  B  < +oo )
32adantl 275 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  < +oo )
4 rexr 7923 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 7930 . . . . . . 7  |- +oo  e.  RR*
6 xrlelttr 9710 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  < +oo )  ->  A  < +oo ) )
75, 6mp3an3 1308 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
84, 7sylan2 284 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
93, 8mpan2d 425 . . . 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 470 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  < +oo )
12 xrrebnd 9723 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1312ad2antrr 480 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
141, 11, 13mpbir2and 929 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 2128   class class class wbr 3965   RRcr 7731   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911    < clt 7912    <_ cle 7913
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-po 4256  df-iso 4257  df-xp 4592  df-cnv 4594  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918
This theorem is referenced by:  xrrege0  9729  tgioo  12957
  Copyright terms: Public domain W3C validator