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Theorem elico1 11447
Description: Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
elico1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )

Proof of Theorem elico1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 11410 . 2  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
21elixx1 11413 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 1758   class class class wbr 4393  (class class class)co 6193   RR*cxr 9521    < clt 9522    <_ cle 9523   [,)cico 11406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-xr 9526  df-ico 11410
This theorem is referenced by:  lbico1  11454  elico2  11463  icodisj  11520  leordtvallem2  18940  pnfnei  18949  mnfnei  18950  metustexhalfOLD  20263  metustexhalf  20264  blval2  20275  metuel2  20279  iscfil2  20902  eliccelico  26205  elicoelioo  26206  xrdifh  26208  fsumrp0cl  26296  xrge0iifcnv  26501  esumpcvgval  26665  tan2h  28565  iocinico  29728  rfcnpre3  29896  bj-flbi3  32836
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