MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mreuni Structured version   Unicode version

Theorem mreuni 13827
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 13821 . 2  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2 mresspw 13819 . 2  |-  ( C  e.  (Moore `  X
)  ->  C  C_  ~P X )
3 elpwuni 4180 . . 3  |-  ( X  e.  C  ->  ( C  C_  ~P X  <->  U. C  =  X ) )
43biimpa 472 . 2  |-  ( ( X  e.  C  /\  C  C_  ~P X )  ->  U. C  =  X )
51, 2, 4syl2anc 644 1  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5456  Moorecmre 13809
This theorem is referenced by:  mreunirn  13828  mrcfval  13835  mrcssv  13841  mrisval  13857  mrelatlub  14614  mreclat  14615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-mre 13813
  Copyright terms: Public domain W3C validator