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Theorem iocssioo 26061
Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
Assertion
Ref Expression
iocssioo  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )

Proof of Theorem iocssioo
Dummy variables  a 
b  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 11304 . 2  |-  (,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <  b ) } )
2 df-ioc 11305 . 2  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <_  b ) } )
3 xrlelttr 11130 . 2  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <  w )  ->  A  <  w
) )
4 xrlelttr 11130 . 2  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <  B )  ->  w  <  B
) )
51, 2, 3, 4ixxss12 11320 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756    C_ wss 3328   class class class wbr 4292  (class class class)co 6091   RR*cxr 9417    < clt 9418    <_ cle 9419   (,)cioo 11300   (,]cioc 11301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-ioo 11304  df-ioc 11305
This theorem is referenced by:  xrge0iifcnv  26363  xrge0iifiso  26365  xrge0iifhom  26367
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