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Theorem mreuni 13502
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 13496 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2 mresspw 13494 . 2  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
3 elpwuni 3989 . . 3  |-  ( X  e.  C  ->  ( C  C_  ~P X  <->  U. C  =  X ) )
43biimpa 470 . 2  |-  ( ( X  e.  C  /\  C  C_  ~P X )  ->  U. C  =  X )
51, 2, 4syl2anc 642 1  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` cfv 5255  Moorecmre 13484
This theorem is referenced by:  mreunirn  13503  mrcfval  13510  mrcssv  13516  mrisval  13532  mrelatlub  14289  mreclat  14290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488
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