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Theorem iccss2 10736
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, 28-Apr-2015.)
Assertion
Ref Expression
iccss2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )

Proof of Theorem iccss2
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 10679 . . . . . 6  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx3g 10685 . . . . 5  |-  ( C  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <_  B ) ) )
32simplbi 446 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
43adantr 451 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* ) )
54simp1d 967 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  e.  RR* )
64simp2d 968 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  B  e.  RR* )
72simprbi 450 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  <_  C  /\  C  <_  B ) )
87adantr 451 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  <_  C  /\  C  <_  B ) )
98simpld 445 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  <_  C )
101elixx3g 10685 . . . . 5  |-  ( D  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <_  D  /\  D  <_  B ) ) )
1110simprbi 450 . . . 4  |-  ( D  e.  ( A [,] B )  ->  ( A  <_  D  /\  D  <_  B ) )
1211simprd 449 . . 3  |-  ( D  e.  ( A [,] B )  ->  D  <_  B )
1312adantl 452 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  D  <_  B )
14 xrletr 10505 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  w )  ->  A  <_  w
) )
15 xrletr 10505 . . 3  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <_  B )  ->  w  <_  B
) )
161, 1, 14, 15ixxss12 10692 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
175, 6, 9, 13, 16syl22anc 1183 1  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   RR*cxr 8882    <_ cle 8884   [,]cicc 10675
This theorem is referenced by:  ordtresticc  16969  iccconn  18351  icccvx  18464  oprpiece1res1  18465  oprpiece1res2  18466  pcoass  18538  dvlip  19356  c1liplem1  19359  dvgt0lem1  19365  ftc2ditglem  19408  unitssxrge0  23299  xrge0iifhmeo  23333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679
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