Theorem intid 4154

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 4117	. . 3 |- { A } e. _V
3		eleq2 2204	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
4	1	snid 3563	. . . 4 |- A e. { A }
5	3, 4	intmin3 3806	. . 3 |- ( { A } e. _V -> |^| { x | A e. x } C_ { A } )
6	2, 5	ax-mp 5	. 2 |- |^| { x | A e. x } C_ { A }
7	1	elintab 3790	. . . 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 1430	. . 3 |- A e. |^| { x | A e. x }
10		snssi 3672	. . 3 |- ( A e. |^| { x | A e. x } -> { A } C_ |^| { x | A e. x } )
11	9, 10	ax-mp 5	. 2 |- { A } C_ |^| { x | A e. x }
12	6, 11	eqssi 3118	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689   C_ wss 3076   {csn 3532   |^|cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-int 3780

This theorem is referenced by: (None)