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Theorem iocssre 10745
Description: A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
iocssre  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )

Proof of Theorem iocssre
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elioc2 10729 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
21biimp3a 1281 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  (
x  e.  RR  /\  A  <  x  /\  x  <_  B ) )
32simp1d 967 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  x  e.  RR )
433expia 1153 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  ->  x  e.  RR )
)
54ssrdv 3198 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   RRcr 8752   RR*cxr 8882    < clt 8883    <_ cle 8884   (,]cioc 10673
This theorem is referenced by:  iocmnfcld  18294  lhop1  19377  negpitopissre  19918  eff1o  19927  dvlog2lem  20015  icccon3  25804
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-ioc 10677
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