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Theorem refrel 19987
Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
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 19984 . 2  |-  Ref  =  { <. x ,  y
>.  |  ( U. y  =  U. x  /\  A. z  e.  x  E. w  e.  y 
z  C_  w ) }
21relopabi 5118 1  |-  Rel  Ref
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1383   A.wral 2793   E.wrex 2794    C_ wss 3461   U.cuni 4234   Rel wrel 4994   Refcref 19981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-xp 4995  df-rel 4996  df-ref 19984
This theorem is referenced by:  isref  19988  refbas  19989  refssex  19990  reftr  19993  refun0  19994  locfinref  27822  refssfne  30152
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