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Theorem xrre3 8836
Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
xrre3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  e.  RR )

Proof of Theorem xrre3
StepHypRef Expression
1 mnflt 8805 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
21adantl 266 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  <  B )
3 mnfxr 8795 . . . . . . 7  |- -oo  e.  RR*
43a1i 9 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  e.  RR* )
5 rexr 7130 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  RR* )
65adantl 266 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  e.  RR* )
7 simpl 106 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  A  e.  RR* )
8 xrltletr 8824 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  A  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
94, 6, 7, 8syl3anc 1146 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
102, 9mpand 413 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( B  <_  A  -> -oo  <  A ) )
1110imp 119 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  B  <_  A
)  -> -oo  <  A
)
1211adantrr 456 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  -> -oo  <  A )
13 simprr 492 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  < +oo )
14 xrrebnd 8833 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1514ad2antrr 465 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1612, 13, 15mpbir2and 862 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    e. wcel 1409   class class class wbr 3792   RRcr 6946   +oocpnf 7116   -oocmnf 7117   RR*cxr 7118    < clt 7119    <_ cle 7120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-po 4061  df-iso 4062  df-xp 4379  df-cnv 4381  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125
This theorem is referenced by: (None)
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