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Theorem elioc2 8906
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 7130 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elioc1 8892 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
31, 2sylan2 274 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )
4 mnfxr 8795 . . . . . . . 8  |- -oo  e.  RR*
54a1i 9 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  e.  RR* )
6 simpll 489 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  e.  RR* )
7 simpr1 921 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR* )
8 mnfle 8814 . . . . . . . 8  |-  ( A  e.  RR*  -> -oo  <_  A )
98ad2antrr 465 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  <_  A )
10 simpr2 922 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  A  <  C )
115, 6, 7, 9, 10xrlelttrd 8827 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> -oo  <  C )
121ad2antlr 466 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  e.  RR* )
13 pnfxr 8793 . . . . . . . 8  |- +oo  e.  RR*
1413a1i 9 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  -> +oo  e.  RR* )
15 simpr3 923 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  <_  B )
16 ltpnf 8803 . . . . . . . 8  |-  ( B  e.  RR  ->  B  < +oo )
1716ad2antlr 466 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  B  < +oo )
187, 12, 14, 15, 17xrlelttrd 8827 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  < +oo )
19 xrrebnd 8833 . . . . . . 7  |-  ( C  e.  RR*  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
207, 19syl 14 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
2111, 18, 20mpbir2and 862 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  C  e.  RR )
2221, 10, 153jca 1095 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( C  e. 
RR*  /\  A  <  C  /\  C  <_  B
) )  ->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )
2322ex 112 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  -> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
24 rexr 7130 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1101 . . 3  |-  ( ( C  e.  RR  /\  A  <  C  /\  C  <_  B )  ->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) )
2623, 25impbid1 134 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( C  e.  RR*  /\  A  <  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
273, 26bitrd 181 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    /\ wa 101    <-> wb 102    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   RRcr 6946   +oocpnf 7116   -oocmnf 7117   RR*cxr 7118    < clt 7119    <_ cle 7120   (,]cioc 8859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-po 4061  df-iso 4062  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-ioc 8863
This theorem is referenced by:  iocssre  8923
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