ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iocssre Unicode version

Theorem iocssre 9765
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 9748 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
21biimp3a 1324 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  (
x  e.  RR  /\  A  <  x  /\  x  <_  B ) )
32simp1d 994 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  x  e.  RR )
433expia 1184 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  ->  x  e.  RR )
)
54ssrdv 3107 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    e. wcel 1481    C_ wss 3075   class class class wbr 3936  (class class class)co 5781   RRcr 7642   RR*cxr 7822    < clt 7823    <_ cle 7824   (,]cioc 9701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-ioc 9705
This theorem is referenced by:  negpitopissre  12982
  Copyright terms: Public domain W3C validator