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Theorem fneref 26055
Description: Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
Assertion
Ref Expression
fneref  |-  ( A  e.  V  ->  A Fne A )

Proof of Theorem fneref
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . 3  |-  U. A  =  U. A
2 ssid 3310 . . . . 5  |-  x  C_  x
3 elequ2 1722 . . . . . . 7  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
4 sseq1 3312 . . . . . . 7  |-  ( z  =  x  ->  (
z  C_  x  <->  x  C_  x
) )
53, 4anbi12d 692 . . . . . 6  |-  ( z  =  x  ->  (
( y  e.  z  /\  z  C_  x
)  <->  ( y  e.  x  /\  x  C_  x ) ) )
65rspcev 2995 . . . . 5  |-  ( ( x  e.  A  /\  ( y  e.  x  /\  x  C_  x ) )  ->  E. z  e.  A  ( y  e.  z  /\  z  C_  x ) )
72, 6mpanr2 666 . . . 4  |-  ( ( x  e.  A  /\  y  e.  x )  ->  E. z  e.  A  ( y  e.  z  /\  z  C_  x
) )
87rgen2 2745 . . 3  |-  A. x  e.  A  A. y  e.  x  E. z  e.  A  ( y  e.  z  /\  z  C_  x )
91, 8pm3.2i 442 . 2  |-  ( U. A  =  U. A  /\  A. x  e.  A  A. y  e.  x  E. z  e.  A  (
y  e.  z  /\  z  C_  x ) )
101, 1isfne2 26042 . 2  |-  ( A  e.  V  ->  ( A Fne A  <->  ( U. A  =  U. A  /\  A. x  e.  A  A. y  e.  x  E. z  e.  A  (
y  e.  z  /\  z  C_  x ) ) ) )
119, 10mpbiri 225 1  |-  ( A  e.  V  ->  A Fne A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    C_ wss 3263   U.cuni 3957   class class class wbr 4153   Fnecfne 26030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-topgen 13594  df-fne 26034
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