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Theorem mreintcl 13513
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 4190 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  ~P C  <->  S  C_  C
) )
21biimpar 471 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  S  e.  ~P C )
323adant3 975 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  e. 
~P C )
4 ismre 13508 . . . 4  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
54simp3bi 972 . . 3  |-  ( C  e.  (Moore `  X
)  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
653ad2ant1 976 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
7 simp3 957 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
8 neeq1 2467 . . . . 5  |-  ( s  =  S  ->  (
s  =/=  (/)  <->  S  =/=  (/) ) )
9 inteq 3881 . . . . . 6  |-  ( s  =  S  ->  |^| s  =  |^| S )
109eleq1d 2362 . . . . 5  |-  ( s  =  S  ->  ( |^| s  e.  C  <->  |^| S  e.  C ) )
118, 10imbi12d 311 . . . 4  |-  ( s  =  S  ->  (
( s  =/=  (/)  ->  |^| s  e.  C )  <->  ( S  =/=  (/)  ->  |^| S  e.  C ) ) )
1211rspcva 2895 . . 3  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  ( S  =/=  (/)  ->  |^| S  e.  C ) )
13123impia 1148 . 2  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
)  /\  S  =/=  (/) )  ->  |^| S  e.  C )
143, 6, 7, 13syl3anc 1182 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271  Moorecmre 13500
This theorem is referenced by:  mreiincl  13514  mrerintcl  13515  mreincl  13517  mremre  13522  submre  13523  mrcflem  13524  mrelatglb  14303  mreclat  14306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-mre 13504
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