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Theorem refrel 26381
Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
refrel  |-  Rel  Ref

Proof of Theorem refrel
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ref 26367 . 2  |-  Ref  =  { <. x ,  y
>.  |  ( U. x  =  U. y  /\  A. z  e.  y  E. w  e.  x  z  C_  w ) }
21relopabi 4827 1  |-  Rel  Ref
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   Rel wrel 4710   Refcref 26363
This theorem is referenced by:  isref  26382  refbas  26383  refssex  26384  reftr  26392  refssfne  26397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712  df-ref 26367
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