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Theorem fneref 26284
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 2283 . . 3  |-  U. A  =  U. A
2 ssid 3197 . . . . 5  |-  x  C_  x
3 elequ2 1689 . . . . . . 7  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
4 sseq1 3199 . . . . . . 7  |-  ( z  =  x  ->  (
z  C_  x  <->  x  C_  x
) )
53, 4anbi12d 691 . . . . . 6  |-  ( z  =  x  ->  (
( y  e.  z  /\  z  C_  x
)  <->  ( y  e.  x  /\  x  C_  x ) ) )
65rspcev 2884 . . . . 5  |-  ( ( x  e.  A  /\  ( y  e.  x  /\  x  C_  x ) )  ->  E. z  e.  A  ( y  e.  z  /\  z  C_  x ) )
72, 6mpanr2 665 . . . 4  |-  ( ( x  e.  A  /\  y  e.  x )  ->  E. z  e.  A  ( y  e.  z  /\  z  C_  x
) )
87rgen2 2639 . . 3  |-  A. x  e.  A  A. y  e.  x  E. z  e.  A  ( y  e.  z  /\  z  C_  x )
91, 8pm3.2i 441 . 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 26271 . 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 224 1  |-  ( A  e.  V  ->  A Fne A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Fnecfne 26259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-fne 26263
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