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Theorem intrr 25111
Description: An interval is a part of  RR. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
intrr  |-  ( I  e.  Intvl  ->  I  C_  RR )

Proof of Theorem intrr
StepHypRef Expression
1 elin 3358 . . 3  |-  ( I  e.  ( ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  i^i  ~P RR )  <-> 
( I  e.  ( ran  (,)  u.  ( ran  (,]  u.  ( ran 
[,)  u.  ran  [,] )
) )  /\  I  e.  ~P RR ) )
2 elpwi 3633 . . . 4  |-  ( I  e.  ~P RR  ->  I 
C_  RR )
32adantl 452 . . 3  |-  ( ( I  e.  ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  /\  I  e.  ~P RR )  ->  I  C_  RR )
41, 3sylbi 187 . 2  |-  ( I  e.  ( ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  i^i  ~P RR )  ->  I  C_  RR )
5 df-intvl 25109 . 2  |-  Intvl  =  ( ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  i^i 
~P RR )
64, 5eleq2s 2375 1  |-  ( I  e.  Intvl  ->  I  C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ran crn 4690   RRcr 8736   (,)cioo 10656   (,]cioc 10657   [,)cico 10658   [,]cicc 10659   Intvlcintvl 25108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-intvl 25109
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