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Theorem mreintcl 13497
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 4174 . . . 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 13492 . . . 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 2454 . . . . 5  |-  ( s  =  S  ->  (
s  =/=  (/)  <->  S  =/=  (/) ) )
9 inteq 3865 . . . . . 6  |-  ( s  =  S  ->  |^| s  =  |^| S )
109eleq1d 2349 . . . . 5  |-  ( s  =  S  ->  ( |^| s  e.  C  <->  |^| S  e.  C ) )
118, 10imbi12d 311 . . . 4  |-  ( s  =  S  ->  (
( s  =/=  (/)  ->  |^| s  e.  C )  <->  ( S  =/=  (/)  ->  |^| S  e.  C ) ) )
1211rspcva 2882 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  mreiincl  13498  mrerintcl  13499  mreincl  13501  mremre  13506  submre  13507  mrcflem  13508  mrelatglb  14287  mreclat  14290
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
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