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Theorem elioc2 10729
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 8893 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elioc1 10714 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
31, 2sylan2 460 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
4 mnfxr 10472 . . . . . . . 8  |-  -oo  e.  RR*
54a1i 10 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  -oo  e.  RR* )
6 simpll 730 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  e.  RR* )
7 simpr1 961 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR* )
8 mnfle 10486 . . . . . . . 8  |-  ( A  e.  RR*  ->  -oo  <_  A )
98ad2antrr 706 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  -oo  <_  A )
10 simpr2 962 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  <  C )
115, 6, 7, 9, 10xrlelttrd 10507 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  -oo  <  C )
121ad2antlr 707 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  e.  RR* )
13 pnfxr 10471 . . . . . . . 8  |-  +oo  e.  RR*
1413a1i 10 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  +oo  e.  RR* )
15 simpr3 963 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <_  B )
16 ltpnf 10479 . . . . . . . 8  |-  ( B  e.  RR  ->  B  <  +oo )
1716ad2antlr 707 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  <  +oo )
187, 12, 14, 15, 17xrlelttrd 10507 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <  +oo )
19 xrrebnd 10513 . . . . . . 7  |-  ( C  e.  RR*  ->  ( C  e.  RR  <->  (  -oo  <  C  /\  C  <  +oo ) ) )
207, 19syl 15 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  <->  (  -oo  <  C  /\  C  <  +oo ) ) )
2111, 18, 20mpbir2and 888 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR )
2221, 10, 153jca 1132 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )
2322ex 423 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  -> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
24 rexr 8893 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1138 . . 3  |-  ( ( C  e.  RR  /\  A  <  C  /\  C  <_  B )  ->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) )
2623, 25impbid1 194 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
273, 26bitrd 244 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   (,]cioc 10673
This theorem is referenced by:  iocssre  10745  ef01bndlem  12480  sin01bnd  12481  cos01bnd  12482  cos1bnd  12483  sinltx  12485  sin01gt0  12486  cos01gt0  12487  sin02gt0  12488  sincos1sgn  12489  sincos2sgn  12490  icoopnst  18453  iocopnst  18454  ismbf3d  19025  aaliou3lem2  19739  aaliou3lem3  19740  pilem2  19844  sinhalfpilem  19850  sincosq1lem  19881  coseq0negpitopi  19887  tangtx  19889  sincos4thpi  19897  efif1olem1  19920  efif1olem2  19921  efif1o  19924  efifo  19925  ellogrn  19933  logimclad  19946  ellogdm  20002  logdmnrp  20004  dvloglem  20011  dvlog2lem  20015  asinneg  20198  atans2  20243  ressatans  20246  abvcxp  20780  ostth2  20802  xrge0iifcv  23331  xrge0iifiso  23332  xrge0iifhom  23334  sinccvglem  24020  dvreasin  25026  areacirclem5  25032  icccon3  25804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioc 10677
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