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Theorem mreuni 17645
Description: Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mreuni (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)

Proof of Theorem mreuni
StepHypRef Expression
1 mre1cl 17639 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
2 mresspw 17637 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3 elpwuni 5110 . . 3 (𝑋𝐶 → (𝐶 ⊆ 𝒫 𝑋 𝐶 = 𝑋))
43biimpa 476 . 2 ((𝑋𝐶𝐶 ⊆ 𝒫 𝑋) → 𝐶 = 𝑋)
51, 2, 4syl2anc 584 1 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605   cuni 4912  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:  mreunirn  17646  mrcfval  17653  mrcssv  17659  mrisval  17675  mrelatlub  18620  mreclatBAD  18621  mreuniss  48696  clduni  48697  mreclat  48786
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