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Theorem refref 26367
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 2438 . . 3  |-  U. A  =  U. A
2 ssid 3369 . . . . 5  |-  x  C_  x
3 sseq2 3372 . . . . . 6  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
43rspcev 3054 . . . . 5  |-  ( ( x  e.  A  /\  x  C_  x )  ->  E. y  e.  A  x  C_  y )
52, 4mpan2 654 . . . 4  |-  ( x  e.  A  ->  E. y  e.  A  x  C_  y
)
65rgen 2773 . . 3  |-  A. x  e.  A  E. y  e.  A  x  C_  y
71, 6pm3.2i 443 . 2  |-  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y )
81, 1isref 26361 . 2  |-  ( A  e.  V  ->  ( A Ref A  <->  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y ) ) )
97, 8mpbiri 226 1  |-  ( A  e.  V  ->  A Ref A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   U.cuni 4017   class class class wbr 4214   Refcref 26342
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  ax-un 4703
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-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-xp 4886  df-rel 4887  df-ref 26346
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