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Theorem mresspw 13494
Description: A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mresspw  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )

Proof of Theorem mresspw
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ismre 13492 . 2  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
21simp1bi 970 1  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  mress  13495  mrerintcl  13499  mreuni  13502  mremre  13506  isacs2  13555  mreacs  13560  isacs3lem  14269  dmdprdd  15237  dprdfeq0  15257  dprdss  15264  dprdz  15265  subgdmdprd  15269  subgdprd  15270  dprd2dlem1  15276  dprd2da  15277  dmdprdsplit2lem  15280  mretopd  16829  ismrc  26776
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-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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