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Theorem elioc2 11587
Description: Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
Assertion
Ref Expression
elioc2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )

Proof of Theorem elioc2
StepHypRef Expression
1 rexr 9639 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elioc1 11571 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
31, 2sylan2 474 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
4 mnfxr 11323 . . . . . . . 8  |- -oo  e.  RR*
54a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  e.  RR* )
6 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  e.  RR* )
7 simpr1 1002 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR* )
8 mnfle 11342 . . . . . . . 8  |-  ( A  e.  RR*  -> -oo  <_  A )
98ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  <_  A )
10 simpr2 1003 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  <  C )
115, 6, 7, 9, 10xrlelttrd 11363 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  <  C )
121ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  e.  RR* )
13 pnfxr 11321 . . . . . . . 8  |- +oo  e.  RR*
1413a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> +oo  e.  RR* )
15 simpr3 1004 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <_  B )
16 ltpnf 11331 . . . . . . . 8  |-  ( B  e.  RR  ->  B  < +oo )
1716ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  < +oo )
187, 12, 14, 15, 17xrlelttrd 11363 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  < +oo )
19 xrrebnd 11369 . . . . . . 7  |-  ( C  e.  RR*  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
207, 19syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
2111, 18, 20mpbir2and 920 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR )
2221, 10, 153jca 1176 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )
2322ex 434 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  -> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
24 rexr 9639 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1182 . . 3  |-  ( ( C  e.  RR  /\  A  <  C  /\  C  <_  B )  ->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) )
2623, 25impbid1 203 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
273, 26bitrd 253 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4447  (class class class)co 6284   RRcr 9491   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629   (,]cioc 11530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-ioc 11534
This theorem is referenced by:  iocssre  11604  ef01bndlem  13780  sin01bnd  13781  cos01bnd  13782  cos1bnd  13783  sinltx  13785  sin01gt0  13786  cos01gt0  13787  sin02gt0  13788  sincos1sgn  13789  sincos2sgn  13790  icoopnst  21202  iocopnst  21203  ismbf3d  21824  aaliou3lem2  22501  aaliou3lem3  22502  pilem2  22609  sinhalfpilem  22617  sincosq1lem  22651  coseq0negpitopi  22657  tangtx  22659  sincos4thpi  22667  efif1olem1  22690  efif1olem2  22691  efif1o  22694  efifo  22695  ellogrn  22703  logimclad  22716  ellogdm  22776  logdmnrp  22778  dvloglem  22785  dvlog2lem  22789  asinneg  22973  atans2  23018  ressatans  23021  abvcxp  23556  ostth2  23578  xrge0iifcv  27580  xrge0iifiso  27581  xrge0iifhom  27583  sinccvglem  28541  dvasin  29708  areacirclem4  29715  gtnelioc  31115  limcicciooub  31207  cncfiooicclem1  31260  fourierdlem4  31439  fourierdlem26  31461  fourierdlem33  31468  fourierdlem37  31472  fourierdlem65  31500  fourierdlem79  31514  fouriersw  31560  bj-pinftyccb  33714  bj-pinftynminfty  33720
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