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Theorem mreuni 16083
 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 16077 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
2 mresspw 16075 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3 elpwuni 4549 . . 3 (𝑋𝐶 → (𝐶 ⊆ 𝒫 𝑋 𝐶 = 𝑋))
43biimpa 500 . 2 ((𝑋𝐶𝐶 ⊆ 𝒫 𝑋) → 𝐶 = 𝑋)
51, 2, 4syl2anc 691 1 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Moorecmre 16065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069 This theorem is referenced by:  mreunirn  16084  mrcfval  16091  mrcssv  16097  mrisval  16113  mrelatlub  17009  mreclatBAD  17010
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