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Theorem iccss 10942
Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
Assertion
Ref Expression
iccss  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  C  /\  D  <_  B
) )  ->  ( C [,] D )  C_  ( A [,] B ) )

Proof of Theorem iccss
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexr 9094 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9094 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
31, 2anim12i 550 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
4 df-icc 10887 . . 3  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
5 xrletr 10712 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  w )  ->  A  <_  w
) )
6 xrletr 10712 . . 3  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <_  B )  ->  w  <_  B
) )
74, 4, 5, 6ixxss12 10900 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
83, 7sylan 458 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  C  /\  D  <_  B
) )  ->  ( C [,] D )  C_  ( A [,] B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    C_ wss 3288   class class class wbr 4180  (class class class)co 6048   RRcr 8953   RR*cxr 9083    <_ cle 9085   [,]cicc 10883
This theorem is referenced by:  xrhmeo  18932  lebnumii  18952  pcoval1  18999  pcoval2  19002  ivthicc  19316  dyaddisjlem  19448  volsup2  19458  volcn  19459  mbfi1fseqlem5  19572  dvcvx  19865  dvfsumle  19866  dvfsumabs  19868  harmonicbnd3  20807  ppisval  20847  chtwordi  20900  ppiwordi  20906  chpub  20965  cvmliftlem2  24934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-pre-lttri 9028  ax-pre-lttrn 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-icc 10887
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