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Theorem iocssre 11595
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 11578 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
21biimp3a 1323 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  (
x  e.  RR  /\  A  <  x  /\  x  <_  B ) )
32simp1d 1003 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  x  e.  RR )
433expia 1193 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  ->  x  e.  RR )
)
54ssrdv 3505 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    e. wcel 1762    C_ wss 3471   class class class wbr 4442  (class class class)co 6277   RRcr 9482   RR*cxr 9618    < clt 9619    <_ cle 9620   (,]cioc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-ioc 11525
This theorem is referenced by:  iocmnfcld  21006  lhop1  22145  negpitopissre  22655  eff1o  22664  dvlog2lem  22756  iocopn  31081  limcicciooub  31136  limcresiooub  31141  cncfiooicclem1  31189  fourierdlem19  31383  fourierdlem33  31397  fourierdlem37  31401  fourierdlem46  31410  fourierdlem48  31412  fourierdlem49  31413  fourierdlem51  31415  fourierdlem63  31427  fourierdlem79  31443  fourierdlem89  31453  fourierdlem90  31454  fourierdlem91  31455  fourierdlem93  31457  fouriersw  31489
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