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Theorem iccssico2 11368
Description: Condition for a closed interval to be a subset of an open 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 11305 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
21elmpt2cl1 6304 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  e.  RR* )
32adantr 465 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  e.  RR* )
41elmpt2cl2 6305 . . 3  |-  ( C  e.  ( A [,) B )  ->  B  e.  RR* )
54adantr 465 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  B  e.  RR* )
61elixx3g 11312 . . . . 5  |-  ( C  e.  ( A [,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <_  C  /\  C  <  B ) ) )
76simprbi 464 . . . 4  |-  ( C  e.  ( A [,) B )  ->  ( A  <_  C  /\  C  <  B ) )
87simpld 459 . . 3  |-  ( C  e.  ( A [,) B )  ->  A  <_  C )
98adantr 465 . 2  |-  ( ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) )  ->  A  <_  C )
101elixx3g 11312 . . . . 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 iccssico 11366 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A [,) B ) )
153, 5, 9, 13, 14syl22anc 1219 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 965    e. wcel 1756   {crab 2718    C_ wss 3327   class class class wbr 4291  (class class class)co 6090   RR*cxr 9416    < clt 9417    <_ cle 9418   [,)cico 11301   [,]cicc 11302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-pre-lttri 9355  ax-pre-lttrn 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-ico 11305  df-icc 11306
This theorem is referenced by:  icopnfhmeo  20514
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