Theorem ifr0 36803

Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ifr0	|- ( _I Fr A <-> A = (/) )

Proof of Theorem ifr0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1855	. . . . 5 |- x = x
2		vex 3048	. . . . . 6 |- x e. _V
3	2	ideq 4987	. . . . 5 |- ( x _I x <-> x = x )
4	1, 3	mpbir 213	. . . 4 |- x _I x
5		frirr 4811	. . . . 5 |- ( ( _I Fr A /\ x e. A ) -> -. x _I x )
6	5	ex 436	. . . 4 |- ( _I Fr A -> ( x e. A -> -. x _I x ) )
7	4, 6	mt2i 122	. . 3 |- ( _I Fr A -> -. x e. A )
8	7	eq0rdv 3769	. 2 |- ( _I Fr A -> A = (/) )
9		fr0 4813	. . 3 |- _I Fr (/)
10		freq2 4805	. . 3 |- ( A = (/) -> ( _I Fr A <-> _I Fr (/) ) )
11	9, 10	mpbiri 237	. 2 |- ( A = (/) -> _I Fr A )
12	8, 11	impbii 191	1 |- ( _I Fr A <-> A = (/) )

Syntax hints:   -. wn 3   <-> wb 188   = wceq 1444   e. wcel 1887   (/)c0 3731   class class class wbr 4402   _I cid 4744   Fr wfr 4790

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-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

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-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-fr 4793  df-xp 4840  df-rel 4841

This theorem is referenced by: (None)