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Theorem elico2 9865
Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
Assertion
Ref Expression
elico2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )

Proof of Theorem elico2
StepHypRef Expression
1 rexr 7936 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 elico1 9851 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
31, 2sylan 281 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
4 mnfxr 7947 . . . . . . . 8  |- -oo  e.  RR*
54a1i 9 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  e.  RR* )
61ad2antrr 480 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  e.  RR* )
7 simpr1 992 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR* )
8 mnflt 9711 . . . . . . . 8  |-  ( A  e.  RR  -> -oo  <  A )
98ad2antrr 480 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  A )
10 simpr2 993 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  <_  C
)
115, 6, 7, 9, 10xrltletrd 9739 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  C )
12 simplr 520 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  e.  RR* )
13 pnfxr 7943 . . . . . . . 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 994 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  <  B
)
16 pnfge 9717 . . . . . . . 8  |-  ( B  e.  RR*  ->  B  <_ +oo )
1716ad2antlr 481 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  <_ +oo )
187, 12, 14, 15, 17xrltletrd 9739 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  < +oo )
19 xrrebnd 9747 . . . . . . 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 933 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR )
2221, 10, 153jca 1166 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) )
2322ex 114 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
24 rexr 7936 . . . 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 141 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
273, 26bitrd 187 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 103    <-> wb 104    /\ w3a 967    e. wcel 2135   class class class wbr 3977  (class class class)co 5837   RRcr 7744   +oocpnf 7922   -oocmnf 7923   RR*cxr 7924    < clt 7925    <_ cle 7926   [,)cico 9818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-po 4269  df-iso 4270  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-iota 5148  df-fun 5185  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-ico 9822
This theorem is referenced by:  icossre  9882  elicopnf  9897  icoshft  9918  modqelico  10260  mulqaddmodid  10290  modqmuladdim  10293  addmodid  10298  icodiamlt  11109  fprodge0  11565  fprodge1  11567  cnbl0  13092  cosq34lt1  13329  cos02pilt1  13330
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