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Theorem mreintcl 15209
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 4557 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  ~P C  <->  S  C_  C
) )
21biimpar 483 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  S  e.  ~P C )
323adant3 1017 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  e. 
~P C )
4 ismre 15204 . . . 4  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
54simp3bi 1014 . . 3  |-  ( C  e.  (Moore `  X
)  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
653ad2ant1 1018 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
7 simp3 999 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
8 neeq1 2684 . . . . 5  |-  ( s  =  S  ->  (
s  =/=  (/)  <->  S  =/=  (/) ) )
9 inteq 4230 . . . . . 6  |-  ( s  =  S  ->  |^| s  =  |^| S )
109eleq1d 2471 . . . . 5  |-  ( s  =  S  ->  ( |^| s  e.  C  <->  |^| S  e.  C ) )
118, 10imbi12d 318 . . . 4  |-  ( s  =  S  ->  (
( s  =/=  (/)  ->  |^| s  e.  C )  <->  ( S  =/=  (/)  ->  |^| S  e.  C ) ) )
1211rspcva 3158 . . 3  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  ( S  =/=  (/)  ->  |^| S  e.  C ) )
13123impia 1194 . 2  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
)  /\  S  =/=  (/) )  ->  |^| S  e.  C )
143, 6, 7, 13syl3anc 1230 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   |^|cint 4227   ` cfv 5569  Moorecmre 15196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-mre 15200
This theorem is referenced by:  mreiincl  15210  mrerintcl  15211  mreincl  15213  mremre  15218  submre  15219  mrcflem  15220  mrelatglb  16138  mreclatBAD  16141
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