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Theorem iccss2 11591
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 11532 . . . . . 6  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx3g 11538 . . . . 5  |-  ( C  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <_  B ) ) )
32simplbi 460 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
43adantr 465 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* ) )
54simp1d 1008 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  e.  RR* )
64simp2d 1009 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  B  e.  RR* )
72simprbi 464 . . . 4  |-  ( C  e.  ( A [,] B )  ->  ( A  <_  C  /\  C  <_  B ) )
87adantr 465 . . 3  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( A  <_  C  /\  C  <_  B ) )
98simpld 459 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  A  <_  C )
101elixx3g 11538 . . . . 5  |-  ( D  e.  ( A [,] B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <_  D  /\  D  <_  B ) ) )
1110simprbi 464 . . . 4  |-  ( D  e.  ( A [,] B )  ->  ( A  <_  D  /\  D  <_  B ) )
1211simprd 463 . . 3  |-  ( D  e.  ( A [,] B )  ->  D  <_  B )
1312adantl 466 . 2  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  ->  D  <_  B )
14 xrletr 11357 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  w )  ->  A  <_  w
) )
15 xrletr 11357 . . 3  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  D  /\  D  <_  B )  ->  w  <_  B
) )
161, 1, 14, 15ixxss12 11545 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
175, 6, 9, 13, 16syl22anc 1229 1  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767    C_ wss 3476   class class class wbr 4447  (class class class)co 6282   RR*cxr 9623    <_ cle 9625   [,]cicc 11528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-icc 11532
This theorem is referenced by:  ordtresticc  19487  iccconn  21067  icccvx  21182  oprpiece1res1  21183  oprpiece1res2  21184  pcoass  21256  dvlip  22126  c1liplem1  22129  dvgt0lem1  22135  ftc2ditglem  22178  ttgcontlem1  23861  unitssxrge0  27515  xrge0iifhmeo  27551
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