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Theorem mreuni 14658
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  e.  (Moore `  X
)  ->  U. C  =  X )

Proof of Theorem mreuni
StepHypRef Expression
1 mre1cl 14652 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2 mresspw 14650 . 2  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
3 elpwuni 4367 . . 3  |-  ( X  e.  C  ->  ( C  C_  ~P X  <->  U. C  =  X ) )
43biimpa 484 . 2  |-  ( ( X  e.  C  /\  C  C_  ~P X )  ->  U. C  =  X )
51, 2, 4syl2anc 661 1  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3437   ~Pcpw 3969   U.cuni 4200   ` cfv 5527  Moorecmre 14640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-mre 14644
This theorem is referenced by:  mreunirn  14659  mrcfval  14666  mrcssv  14672  mrisval  14688  mrelatlub  15476  mreclatBAD  15477
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