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Theorem fneref 26355
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 2435 . . 3  |-  U. A  =  U. A
2 ssid 3359 . . . . 5  |-  x  C_  x
3 elequ2 1730 . . . . . . 7  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
4 sseq1 3361 . . . . . . 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 3044 . . . . 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 2794 . . 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 26342 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   class class class wbr 4204   Fnecfne 26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659  df-fne 26334
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