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Theorem mreuni 17415
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 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐢 = 𝑋)

Proof of Theorem mreuni
StepHypRef Expression
1 mre1cl 17409 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
2 mresspw 17407 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐢 βŠ† 𝒫 𝑋)
3 elpwuni 5064 . . 3 (𝑋 ∈ 𝐢 β†’ (𝐢 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐢 = 𝑋))
43biimpa 478 . 2 ((𝑋 ∈ 𝐢 ∧ 𝐢 βŠ† 𝒫 𝑋) β†’ βˆͺ 𝐢 = 𝑋)
51, 2, 4syl2anc 585 1 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐢 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   βŠ† wss 3909  π’« cpw 4559  βˆͺ cuni 4864  β€˜cfv 6492  Moorecmre 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fv 6500  df-mre 17401
This theorem is referenced by:  mreunirn  17416  mrcfval  17423  mrcssv  17429  mrisval  17445  mrelatlub  18386  mreclatBAD  18387  mreuniss  46687  clduni  46688  mreclat  46777
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