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Theorem elioc2 11356
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 9427 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elioc1 11340 . . 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 11092 . . . . . . . 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 994 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR* )
8 mnfle 11111 . . . . . . . 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 995 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  <  C )
115, 6, 7, 9, 10xrlelttrd 11132 . . . . . 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 11090 . . . . . . . 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 996 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <_  B )
16 ltpnf 11100 . . . . . . . 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 11132 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  < +oo )
19 xrrebnd 11138 . . . . . . 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 913 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR )
2221, 10, 153jca 1168 . . . 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 9427 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1174 . . 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 965    e. wcel 1756   class class class wbr 4290  (class class class)co 6089   RRcr 9279   +oocpnf 9413   -oocmnf 9414   RR*cxr 9415    < clt 9416    <_ cle 9417   (,]cioc 11299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-pre-lttri 9354  ax-pre-lttrn 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-ioc 11303
This theorem is referenced by:  iocssre  11373  ef01bndlem  13466  sin01bnd  13467  cos01bnd  13468  cos1bnd  13469  sinltx  13471  sin01gt0  13472  cos01gt0  13473  sin02gt0  13474  sincos1sgn  13475  sincos2sgn  13476  icoopnst  20509  iocopnst  20510  ismbf3d  21130  aaliou3lem2  21807  aaliou3lem3  21808  pilem2  21915  sinhalfpilem  21923  sincosq1lem  21957  coseq0negpitopi  21963  tangtx  21965  sincos4thpi  21973  efif1olem1  21996  efif1olem2  21997  efif1o  22000  efifo  22001  ellogrn  22009  logimclad  22022  ellogdm  22082  logdmnrp  22084  dvloglem  22091  dvlog2lem  22095  asinneg  22279  atans2  22324  ressatans  22327  abvcxp  22862  ostth2  22884  xrge0iifcv  26362  xrge0iifiso  26363  xrge0iifhom  26365  sinccvglem  27315  dvasin  28477  areacirclem4  28484  bj-pinftyccb  32541  bj-pinftynminfty  32547
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