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Theorem mre1cl 15083
Description: In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mre1cl  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )

Proof of Theorem mre1cl
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ismre 15079 . 2  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
21simp2bi 1010 1  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    =/= wne 2649   A.wral 2804    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   |^|cint 4271   ` cfv 5570  Moorecmre 15071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-mre 15075
This theorem is referenced by:  mrerintcl  15086  mreriincl  15087  mreuni  15089  mremre  15093  mrcflem  15095  mrcval  15099  mrccl  15100  mrcun  15111  mrelatglb0  16014  mreclatBAD  16016  mretopd  19760
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