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Theorem xrre3 11230
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 11191 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
21adantl 466 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  <  B )
3 mnfxr 11181 . . . . . . 7  |- -oo  e.  RR*
43a1i 11 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  e.  RR* )
5 rexr 9516 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  RR* )
65adantl 466 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  e.  RR* )
7 simpl 457 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  A  e.  RR* )
8 xrltletr 11218 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  A  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
94, 6, 7, 8syl3anc 1219 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
102, 9mpand 675 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( B  <_  A  -> -oo  <  A ) )
1110imp 429 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  B  <_  A
)  -> -oo  <  A
)
1211adantrr 716 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  -> -oo  <  A )
13 simprr 756 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  < +oo )
14 xrrebnd 11227 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1514ad2antrr 725 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1612, 13, 15mpbir2and 913 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1757   class class class wbr 4376   RRcr 9368   +oocpnf 9502   -oocmnf 9503   RR*cxr 9504    < clt 9505    <_ cle 9506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-pre-lttri 9443  ax-pre-lttrn 9444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-po 4725  df-so 4726  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511
This theorem is referenced by:  sibfinima  26845  orvcgteel  26970  ismblfin  28556
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