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Theorem refref 19992
Description: Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
refref  |-  ( A  e.  V  ->  A Ref A )

Proof of Theorem refref
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . 3  |-  U. A  =  U. A
2 ssid 3508 . . . . 5  |-  x  C_  x
3 sseq2 3511 . . . . . 6  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
43rspcev 3196 . . . . 5  |-  ( ( x  e.  A  /\  x  C_  x )  ->  E. y  e.  A  x  C_  y )
52, 4mpan2 671 . . . 4  |-  ( x  e.  A  ->  E. y  e.  A  x  C_  y
)
65rgen 2803 . . 3  |-  A. x  e.  A  E. y  e.  A  x  C_  y
71, 6pm3.2i 455 . 2  |-  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y )
81, 1isref 19988 . 2  |-  ( A  e.  V  ->  ( A Ref A  <->  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y ) ) )
97, 8mpbiri 233 1  |-  ( A  e.  V  ->  A Ref A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   U.cuni 4234   class class class wbr 4437   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-8 1806  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-pow 4615  ax-pr 4676  ax-un 6577
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-eu 2272  df-mo 2273  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-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-ref 19984
This theorem is referenced by:  locfinref  27822
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