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Theorem mreuni 16188
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 16182 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
2 mresspw 16180 . 2 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3 elpwuni 4584 . . 3 (𝑋𝐶 → (𝐶 ⊆ 𝒫 𝑋 𝐶 = 𝑋))
43biimpa 501 . 2 ((𝑋𝐶𝐶 ⊆ 𝒫 𝑋) → 𝐶 = 𝑋)
51, 2, 4syl2anc 692 1 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wss 3559  𝒫 cpw 4135   cuni 4407  cfv 5852  Moorecmre 16170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-mre 16174
This theorem is referenced by:  mreunirn  16189  mrcfval  16196  mrcssv  16202  mrisval  16218  mrelatlub  17114  mreclatBAD  17115
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