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Theorem mresspw 13587
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 13585 . 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 1710    =/= wne 2521   A.wral 2619    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   |^|cint 3941   ` cfv 5334  Moorecmre 13577
This theorem is referenced by:  mress  13588  mrerintcl  13592  mreuni  13595  mremre  13599  isacs2  13648  mreacs  13653  isacs3lem  14362  dmdprdd  15330  dprdfeq0  15350  dprdss  15357  dprdz  15358  subgdmdprd  15362  subgdprd  15363  dprd2dlem1  15369  dprd2da  15370  dmdprdsplit2lem  15373  mretopd  16929  ismrc  26099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-mre 13581
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