Theorem fneref 32345

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. A = U. A
2		ssid 3624	. . . . 5 |- x C_ x
3		elequ2 2004	. . . . . . 7 |- ( z = x -> ( y e. z <-> y e. x ) )
4		sseq1 3626	. . . . . . 7 |- ( z = x -> ( z C_ x <-> x C_ x ) )
5	3, 4	anbi12d 747	. . . . . 6 |- ( z = x -> ( ( y e. z /\ z C_ x ) <-> ( y e. x /\ x C_ x ) ) )
6	5	rspcev 3309	. . . . 5 |- ( ( x e. A /\ ( y e. x /\ x C_ x ) ) -> E. z e. A ( y e. z /\ z C_ x ) )
7	2, 6	mpanr2 720	. . . 4 |- ( ( x e. A /\ y e. x ) -> E. z e. A ( y e. z /\ z C_ x ) )
8	7	rgen2 2975	. . 3 |- A. x e. A A. y e. x E. z e. A ( y e. z /\ z C_ x )
9	1, 8	pm3.2i 471	. 2 |- ( U. A = U. A /\ A. x e. A A. y e. x E. z e. A ( y e. z /\ z C_ x ) )
10	1, 1	isfne2 32337	. 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 ) ) ) )
11	9, 10	mpbiri 248	1 |- ( A e. V -> A Fne A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   class class class wbr 4653   Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-fne 32332

This theorem is referenced by: (None)