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Theorem xrre3 11293
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 11254 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
21adantl 464 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  <  B )
3 mnfxr 11244 . . . . . . 7  |- -oo  e.  RR*
43a1i 11 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  -> -oo  e.  RR* )
5 rexr 9550 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  RR* )
65adantl 464 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  e.  RR* )
7 simpl 455 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  A  e.  RR* )
8 xrltletr 11281 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  A  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
94, 6, 7, 8syl3anc 1226 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( -oo  <  B  /\  B  <_  A )  -> -oo  <  A ) )
102, 9mpand 673 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( B  <_  A  -> -oo  <  A ) )
1110imp 427 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  B  <_  A
)  -> -oo  <  A
)
1211adantrr 714 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  -> -oo  <  A )
13 simprr 755 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  A  < +oo )
14 xrrebnd 11290 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1514ad2antrr 723 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  < +oo ) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1612, 13, 15mpbir2and 920 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 367    e. wcel 1826   class class class wbr 4367   RRcr 9402   +oocpnf 9536   -oocmnf 9537   RR*cxr 9538    < clt 9539    <_ cle 9540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-pre-lttri 9477  ax-pre-lttrn 9478
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545
This theorem is referenced by:  sibfinima  28464  orvcgteel  28589  ismblfin  30220  elicore  31703
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