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Theorem mreuni 17548
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 17542 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
2 mresspw 17540 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐢 βŠ† 𝒫 𝑋)
3 elpwuni 5107 . . 3 (𝑋 ∈ 𝐢 β†’ (𝐢 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐢 = 𝑋))
43biimpa 475 . 2 ((𝑋 ∈ 𝐢 ∧ 𝐢 βŠ† 𝒫 𝑋) β†’ βˆͺ 𝐢 = 𝑋)
51, 2, 4syl2anc 582 1 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐢 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  β€˜cfv 6542  Moorecmre 17530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-mre 17534
This theorem is referenced by:  mreunirn  17549  mrcfval  17556  mrcssv  17562  mrisval  17578  mrelatlub  18519  mreclatBAD  18520  mreuniss  47619  clduni  47620  mreclat  47709
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