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Theorem iccss2 9695
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 9646 . . . . . 6  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx3g 9652 . . . . 5  |-  ( C  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <_  B ) ) )
32simplbi 272 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
43adantr 274 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* ) )
54simp1d 978 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  e.  RR* )
64simp2d 979 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  B  e.  RR* )
72simprbi 273 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  <_  C  /\  C  <_  B ) )
87adantr 274 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  <_  C  /\  C  <_  B ) )
98simpld 111 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  <_  C )
101elixx3g 9652 . . . . 5  |-  ( D  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <_  D  /\  D  <_  B ) ) )
1110simprbi 273 . . . 4  |-  ( D  e.  ( A [,] B )  ->  ( A  <_  D  /\  D  <_  B ) )
1211simprd 113 . . 3  |-  ( D  e.  ( A [,] B )  ->  D  <_  B )
1312adantl 275 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  D  <_  B )
14 xrletr 9559 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  w )  ->  A  <_  w
) )
15 xrletr 9559 . . 3  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <_  B )  ->  w  <_  B
) )
161, 1, 14, 15ixxss12 9657 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
175, 6, 9, 13, 16syl22anc 1202 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 103    /\ w3a 947    e. wcel 1465    C_ wss 3041   class class class wbr 3899  (class class class)co 5742   RR*cxr 7767    <_ cle 7769   [,]cicc 9642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-icc 9646
This theorem is referenced by: (None)
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