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Theorem elioc1 10714
Description: Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
elioc1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <_  B ) ) )

Proof of Theorem elioc1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 10677 . 2  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
21elixx1 10681 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   RR*cxr 8882    < clt 8883    <_ cle 8884   (,]cioc 10673
This theorem is referenced by:  ubioc1  10721  elioc2  10729  leordtvallem1  16956  pnfnei  16966  mnfnei  16967  xrge0tsms  18355  lhop1  19377  xrlimcnp  20279  iocinioc2  23287  xrge0iifcnv  23330  lmxrge0  23390  xrge0tsmsd  23397  rfcnpre4  27808
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-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-xr 8887  df-ioc 10677
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