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Theorem mresspw 17637
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 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)

Proof of Theorem mresspw
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ismre 17635 . 2 (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋𝑋𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → 𝑠𝐶)))
21simp1bi 1144 1 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  wral 3059  wss 3963  c0 4339  𝒫 cpw 4605   cint 4951  cfv 6563  Moorecmre 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-mre 17631
This theorem is referenced by:  mress  17638  mrerintcl  17642  mreuni  17645  mremre  17649  isacs2  17698  mreacs  17703  isacs3lem  18600  dmdprdd  20034  dprdfeq0  20057  dprdss  20064  dprdz  20065  subgdmdprd  20069  subgdprd  20070  dprd2dlem1  20076  dprd2da  20077  dmdprdsplit2lem  20080  mretopd  23116  ismrc  42689
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