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Theorem iccssico2 9046
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 8993 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
21elmpt2cl1 5730 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  e.  RR* )
32adantr 270 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  e.  RR* )
41elmpt2cl2 5731 . . 3  |-  ( C  e.  ( A [,) B )  ->  B  e.  RR* )
54adantr 270 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  B  e.  RR* )
61elixx3g 9000 . . . . 5  |-  ( C  e.  ( A [,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <  B ) ) )
76simprbi 269 . . . 4  |-  ( C  e.  ( A [,) B )  ->  ( A  <_  C  /\  C  <  B ) )
87simpld 110 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  <_  C )
98adantr 270 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  <_  C )
101elixx3g 9000 . . . . 5  |-  ( D  e.  ( A [,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <_  D  /\  D  <  B ) ) )
1110simprbi 269 . . . 4  |-  ( D  e.  ( A [,) B )  ->  ( A  <_  D  /\  D  <  B ) )
1211simprd 112 . . 3  |-  ( D  e.  ( A [,) B )  ->  D  <  B )
1312adantl 271 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  D  <  B )
14 iccssico 9044 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A [,) B ) )
153, 5, 9, 13, 14syl22anc 1171 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 102    /\ w3a 920    e. wcel 1434   {crab 2353    C_ wss 2974   class class class wbr 3793  (class class class)co 5543   RR*cxr 7214    < clt 7215    <_ cle 7216   [,)cico 8989   [,]cicc 8990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-ico 8993  df-icc 8994
This theorem is referenced by: (None)
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