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Theorem mreintcl 14839
Description: A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mreintcl  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )

Proof of Theorem mreintcl
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 elpw2g 4603 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  ~P C  <->  S  C_  C
) )
21biimpar 485 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  S  e.  ~P C )
323adant3 1011 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  e. 
~P C )
4 ismre 14834 . . . 4  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
54simp3bi 1008 . . 3  |-  ( C  e.  (Moore `  X
)  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
653ad2ant1 1012 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
7 simp3 993 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
8 neeq1 2741 . . . . 5  |-  ( s  =  S  ->  (
s  =/=  (/)  <->  S  =/=  (/) ) )
9 inteq 4278 . . . . . 6  |-  ( s  =  S  ->  |^| s  =  |^| S )
109eleq1d 2529 . . . . 5  |-  ( s  =  S  ->  ( |^| s  e.  C  <->  |^| S  e.  C ) )
118, 10imbi12d 320 . . . 4  |-  ( s  =  S  ->  (
( s  =/=  (/)  ->  |^| s  e.  C )  <->  ( S  =/=  (/)  ->  |^| S  e.  C ) ) )
1211rspcva 3205 . . 3  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  ( S  =/=  (/)  ->  |^| S  e.  C ) )
13123impia 1188 . 2  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
)  /\  S  =/=  (/) )  ->  |^| S  e.  C )
143, 6, 7, 13syl3anc 1223 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   |^|cint 4275   ` cfv 5579  Moorecmre 14826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-mre 14830
This theorem is referenced by:  mreiincl  14840  mrerintcl  14841  mreincl  14843  mremre  14848  submre  14849  mrcflem  14850  mrelatglb  15660  mreclatBAD  15663
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