Theorem ifr0 36873

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 1863	. . . . 5 |- x = x
2		vex 3034	. . . . . 6 |- x e. _V
3	2	ideq 4992	. . . . 5 |- ( x _I x <-> x = x )
4	1, 3	mpbir 214	. . . 4 |- x _I x
5		frirr 4816	. . . . 5 |- ( ( _I Fr A /\ x e. A ) -> -. x _I x )
6	5	ex 441	. . . 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 3773	. 2 |- ( _I Fr A -> A = (/) )
9		fr0 4818	. . 3 |- _I Fr (/)
10		freq2 4810	. . 3 |- ( A = (/) -> ( _I Fr A <-> _I Fr (/) ) )
11	9, 10	mpbiri 241	. 2 |- ( A = (/) -> _I Fr A )
12	8, 11	impbii 192	1 |- ( _I Fr A <-> A = (/) )

Syntax hints:   -. wn 3   <-> wb 189   = wceq 1452   e. wcel 1904   (/)c0 3722   class class class wbr 4395   _I cid 4749   Fr wfr 4795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-fr 4798  df-xp 4845  df-rel 4846

This theorem is referenced by: (None)