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Theorem mresspw 13822
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 13820 . 2  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
21simp1bi 973 1  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052   ` cfv 5457  Moorecmre 13812
This theorem is referenced by:  mress  13823  mrerintcl  13827  mreuni  13830  mremre  13834  isacs2  13883  mreacs  13888  isacs3lem  14597  dmdprdd  15565  dprdfeq0  15585  dprdss  15592  dprdz  15593  subgdmdprd  15597  subgdprd  15598  dprd2dlem1  15604  dprd2da  15605  dmdprdsplit2lem  15608  mretopd  17161  ismrc  26769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-mre 13816
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