Theorem intid 3979

Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)

Hypothesis

Ref	Expression
intid.1	|- A e. _V

Assertion

Ref	Expression
intid	|- |^| { x | A e. x } = { A }

Distinct variable group:   x, A

Proof of Theorem intid

Step	Hyp	Ref	Expression
1		intid.1	. . . 4 |- A e. _V
2	1	snex 3957	. . 3 |- { A } e. _V
3		eleq2 2142	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
4	1	snid 3425	. . . 4 |- A e. { A }
5	3, 4	intmin3 3663	. . 3 |- ( { A } e. _V -> |^| { x | A e. x } C_ { A } )
6	2, 5	ax-mp 7	. 2 |- |^| { x | A e. x } C_ { A }
7	1	elintab 3647	. . . 4 |- ( A e. |^| { x | A e. x } <-> A. x ( A e. x -> A e. x ) )
8		id 19	. . . 4 |- ( A e. x -> A e. x )
9	7, 8	mpgbir 1382	. . 3 |- A e. |^| { x | A e. x }
10		snssi 3529	. . 3 |- ( A e. |^| { x | A e. x } -> { A } C_ |^| { x | A e. x } )
11	9, 10	ax-mp 7	. 2 |- { A } C_ |^| { x | A e. x }
12	6, 11	eqssi 3015	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973   {csn 3398   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-int 3637

This theorem is referenced by: (None)