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Theorem elico2 11378
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 9448 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 elico1 11362 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
31, 2sylan 471 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
4 mnfxr 11113 . . . . . . . 8  |- -oo  e.  RR*
54a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  e.  RR* )
61ad2antrr 725 . . . . . . 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 mnflt 11123 . . . . . . . 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, 10xrltletrd 11154 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  C )
12 simplr 754 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  e.  RR* )
13 pnfxr 11111 . . . . . . . 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 pnfge 11129 . . . . . . . 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, 17xrltletrd 11154 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  < +oo )
19 xrrebnd 11159 . . . . . . 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 9448 . . . 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 4311  (class class class)co 6110   RRcr 9300   +oocpnf 9434   -oocmnf 9435   RR*cxr 9436    < clt 9437    <_ cle 9438   [,)cico 11321
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 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-pre-lttri 9375  ax-pre-lttrn 9376
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-po 4660  df-so 4661  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-ico 11325
This theorem is referenced by:  icossre  11395  elicopnf  11404  icoshft  11426  rge0srg  17901  metustexhalfOLD  20157  metustexhalf  20158  cnbl0  20372  icoopnst  20530  iocopnst  20531  icopnfcnv  20533  icopnfhmeo  20534  iccpnfcnv  20535  psercnlem2  21908  psercnlem1  21909  psercn  21910  abelth  21925  tanord1  22012  tanord  22013  efopnlem1  22120  logtayl  22124  rlimcnp  22378  rlimcnp2  22379  dchrvmasumlem2  22766  dchrvmasumiflem1  22769  pntlemb  22865  pnt  22882  ubico  26084  xrge0slmod  26331  voliune  26664  volfiniune  26665  dya2icoseg  26711  sibfinima  26744  tan2h  28447  itg2addnclem2  28467  icodiamlt  29184  modelico  29185
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