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Theorem iocssre 11715
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 11698 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
21biimp3a 1364 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  (
x  e.  RR  /\  A  <  x  /\  x  <_  B ) )
32simp1d 1017 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  x  e.  RR )
433expia 1207 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  ->  x  e.  RR )
)
54ssrdv 3470 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1868    C_ wss 3436   class class class wbr 4420  (class class class)co 6302   RRcr 9539   RR*cxr 9675    < clt 9676    <_ cle 9677   (,]cioc 11637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-pre-lttri 9614  ax-pre-lttrn 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-ioc 11641
This theorem is referenced by:  iocmnfcld  21776  lhop1  22953  negpitopissre  23476  eff1o  23485  dvlog2lem  23584  iocopn  37452  limcicciooub  37537  limcresiooub  37543  fourierdlem19  37808  fourierdlem33  37823  fourierdlem37  37827  fourierdlem46  37836  fourierdlem48  37838  fourierdlem49  37839  fourierdlem51  37841  fourierdlem63  37853  fourierdlem79  37869  fourierdlem89  37879  fourierdlem90  37880  fourierdlem91  37881  fourierdlem93  37883  fouriersw  37915
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