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Theorem elioc2 10965
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 9122 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elioc1 10950 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
31, 2sylan2 461 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
4 mnfxr 10706 . . . . . . . 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 731 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  e.  RR* )
7 simpr1 963 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR* )
8 mnfle 10721 . . . . . . . 8  |-  ( A  e.  RR*  ->  -oo  <_  A )
98ad2antrr 707 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  -oo  <_  A )
10 simpr2 964 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  <  C )
115, 6, 7, 9, 10xrlelttrd 10742 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  -oo  <  C )
121ad2antlr 708 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  e.  RR* )
13 pnfxr 10705 . . . . . . . 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 965 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <_  B )
16 ltpnf 10713 . . . . . . . 8  |-  ( B  e.  RR  ->  B  <  +oo )
1716ad2antlr 708 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  <  +oo )
187, 12, 14, 15, 17xrlelttrd 10742 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <  +oo )
19 xrrebnd 10748 . . . . . . 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 889 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR )
2221, 10, 153jca 1134 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )
2322ex 424 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  -> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
24 rexr 9122 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1140 . . 3  |-  ( ( C  e.  RR  /\  A  <  C  /\  C  <_  B )  ->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) )
2623, 25impbid1 195 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
273, 26bitrd 245 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 177    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   (,]cioc 10909
This theorem is referenced by:  iocssre  10982  ef01bndlem  12777  sin01bnd  12778  cos01bnd  12779  cos1bnd  12780  sinltx  12782  sin01gt0  12783  cos01gt0  12784  sin02gt0  12785  sincos1sgn  12786  sincos2sgn  12787  icoopnst  18956  iocopnst  18957  ismbf3d  19538  aaliou3lem2  20252  aaliou3lem3  20253  pilem2  20360  sinhalfpilem  20366  sincosq1lem  20397  coseq0negpitopi  20403  tangtx  20405  sincos4thpi  20413  efif1olem1  20436  efif1olem2  20437  efif1o  20440  efifo  20441  ellogrn  20449  logimclad  20462  ellogdm  20522  logdmnrp  20524  dvloglem  20531  dvlog2lem  20535  asinneg  20718  atans2  20763  ressatans  20766  abvcxp  21301  ostth2  21323  xrge0iifcv  24312  xrge0iifiso  24313  xrge0iifhom  24315  sinccvglem  25101  dvreasin  26270  areacirclem5  26276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ioc 10913
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