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Theorem refref 20528
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 2451 . . 3  |-  U. A  =  U. A
2 ssid 3451 . . . . 5  |-  x  C_  x
3 sseq2 3454 . . . . . 6  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
43rspcev 3150 . . . . 5  |-  ( ( x  e.  A  /\  x  C_  x )  ->  E. y  e.  A  x  C_  y )
52, 4mpan2 677 . . . 4  |-  ( x  e.  A  ->  E. y  e.  A  x  C_  y
)
65rgen 2747 . . 3  |-  A. x  e.  A  E. y  e.  A  x  C_  y
71, 6pm3.2i 457 . 2  |-  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y )
81, 1isref 20524 . 2  |-  ( A  e.  V  ->  ( A Ref A  <->  ( U. A  =  U. A  /\  A. x  e.  A  E. y  e.  A  x  C_  y ) ) )
97, 8mpbiri 237 1  |-  ( A  e.  V  ->  A Ref A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    C_ wss 3404   U.cuni 4198   class class class wbr 4402   Refcref 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-ref 20520
This theorem is referenced by:  locfinref  28668
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