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Theorem iccss 10983
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 9135 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9135 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
31, 2anim12i 551 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
4 df-icc 10928 . . 3  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
5 xrletr 10753 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  w )  ->  A  <_  w
) )
6 xrletr 10753 . . 3  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <_  B )  ->  w  <_  B
) )
74, 4, 5, 6ixxss12 10941 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
83, 7sylan 459 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 360    e. wcel 1726    C_ wss 3322   class class class wbr 4215  (class class class)co 6084   RRcr 8994   RR*cxr 9124    <_ cle 9126   [,]cicc 10924
This theorem is referenced by:  xrhmeo  18976  lebnumii  18996  pcoval1  19043  pcoval2  19046  ivthicc  19360  dyaddisjlem  19492  volsup2  19502  volcn  19503  mbfi1fseqlem5  19614  dvcvx  19909  dvfsumle  19910  dvfsumabs  19912  harmonicbnd3  20851  ppisval  20891  chtwordi  20944  ppiwordi  20950  chpub  21009  cvmliftlem2  24978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-icc 10928
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