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Theorem iccssico2 11708
Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
Assertion
Ref Expression
iccssico2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  -> 
( C [,] D
)  C_  ( A [,) B ) )

Proof of Theorem iccssico2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 11641 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
21elmpt2cl1 6526 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  e.  RR* )
32adantr 466 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  e.  RR* )
41elmpt2cl2 6527 . . 3  |-  ( C  e.  ( A [,) B )  ->  B  e.  RR* )
54adantr 466 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  B  e.  RR* )
61elixx3g 11648 . . . . 5  |-  ( C  e.  ( A [,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <  B ) ) )
76simprbi 465 . . . 4  |-  ( C  e.  ( A [,) B )  ->  ( A  <_  C  /\  C  <  B ) )
87simpld 460 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  <_  C )
98adantr 466 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  <_  C )
101elixx3g 11648 . . . . 5  |-  ( D  e.  ( A [,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <_  D  /\  D  <  B ) ) )
1110simprbi 465 . . . 4  |-  ( D  e.  ( A [,) B )  ->  ( A  <_  D  /\  D  <  B ) )
1211simprd 464 . . 3  |-  ( D  e.  ( A [,) B )  ->  D  <  B )
1312adantl 467 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  D  <  B )
14 iccssico 11706 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A [,) B ) )
153, 5, 9, 13, 14syl22anc 1265 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 370    /\ w3a 982    e. wcel 1870   {crab 2786    C_ wss 3442   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   [,)cico 11637   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-ico 11641  df-icc 11642
This theorem is referenced by:  icopnfhmeo  21867
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