Theorem intid 4345

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 4303	. . 3 |- { A } e. _V
3		eleq2 2298	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
4	1	snid 3725	. . . 4 |- A e. { A }
5	3, 4	intmin3 3981	. . 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 3965	. . . 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 1502	. . 3 |- A e. |^| { x | A e. x }
10		snssi 3843	. . 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 3258	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {cab 2220   _Vcvv 2815   C_ wss 3214   {csn 3694   |^|cint 3954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-int 3955

This theorem is referenced by: (None)