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Theorem elico2 10730
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 8893 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 elico1 10715 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
31, 2sylan 457 . 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* )
61ad2antrr 706 . . . . . . 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 mnflt 10480 . . . . . . . 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, 10xrltletrd 10508 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  -oo  <  C )
12 simplr 731 . . . . . . 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 pnfge 10485 . . . . . . . 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, 17xrltletrd 10508 . . . . . 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   [,)cico 10674
This theorem is referenced by:  icossre  10746  elicopnf  10755  icoshft  10774  cnbl0  18299  icoopnst  18453  iocopnst  18454  icopnfcnv  18456  icopnfhmeo  18457  iccpnfcnv  18458  psercnlem2  19816  psercnlem1  19817  psercn  19818  abelth  19833  tanord1  19915  tanord  19916  efopnlem1  20019  logtayl  20023  rlimcnp  20276  rlimcnp2  20277  dchrvmasumlem2  20663  dchrvmasumiflem1  20666  pntlemb  20762  pnt  20779  ubico  23283  cnre2csqlem  23309  dya2iocseg  23594  itg2addnclem2  25004  icccon2  25803  icodiamlt  27008  modelico  27009
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-ico 10678
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