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Theorem mreuni 16859
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 16853 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
2 mresspw 16851 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3 elpwuni 5018 . . 3 (𝑋𝐶 → (𝐶 ⊆ 𝒫 𝑋 𝐶 = 𝑋))
43biimpa 477 . 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 1528  wcel 2105  wss 3933  𝒫 cpw 4535   cuni 4830  cfv 6348  Moorecmre 16841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-mre 16845
This theorem is referenced by:  mreunirn  16860  mrcfval  16867  mrcssv  16873  mrisval  16889  mrelatlub  17784  mreclatBAD  17785
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