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Theorem mreintcl 13822
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 4365 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  ~P C  <->  S  C_  C
) )
21biimpar 473 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  S  e.  ~P C )
323adant3 978 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  e. 
~P C )
4 ismre 13817 . . . 4  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
54simp3bi 975 . . 3  |-  ( C  e.  (Moore `  X
)  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
653ad2ant1 979 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) )
7 simp3 960 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  S  =/=  (/) )
8 neeq1 2611 . . . . 5  |-  ( s  =  S  ->  (
s  =/=  (/)  <->  S  =/=  (/) ) )
9 inteq 4055 . . . . . 6  |-  ( s  =  S  ->  |^| s  =  |^| S )
109eleq1d 2504 . . . . 5  |-  ( s  =  S  ->  ( |^| s  e.  C  <->  |^| S  e.  C ) )
118, 10imbi12d 313 . . . 4  |-  ( s  =  S  ->  (
( s  =/=  (/)  ->  |^| s  e.  C )  <->  ( S  =/=  (/)  ->  |^| S  e.  C ) ) )
1211rspcva 3052 . . 3  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )  ->  ( S  =/=  (/)  ->  |^| S  e.  C ) )
13123impia 1151 . 2  |-  ( ( S  e.  ~P C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
)  /\  S  =/=  (/) )  ->  |^| S  e.  C )
143, 6, 7, 13syl3anc 1185 1  |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/=  (/) )  ->  |^| S  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052   ` cfv 5456  Moorecmre 13809
This theorem is referenced by:  mreiincl  13823  mrerintcl  13824  mreincl  13826  mremre  13831  submre  13832  mrcflem  13833  mrelatglb  14612  mreclat  14615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-mre 13813
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