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Theorem iocssre 10982
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 10965 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
21biimp3a 1283 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  (
x  e.  RR  /\  A  <  x  /\  x  <_  B ) )
32simp1d 969 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR  /\  x  e.  ( A (,] B
) )  ->  x  e.  RR )
433expia 1155 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  ->  x  e.  RR )
)
54ssrdv 3346 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    C_ wss 3312   class class class wbr 4204  (class class class)co 6073   RRcr 8981   RR*cxr 9111    < clt 9112    <_ cle 9113   (,]cioc 10909
This theorem is referenced by:  iocmnfcld  18795  lhop1  19890  negpitopissre  20434  eff1o  20443  dvlog2lem  20535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ioc 10913
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