Theorem intid 4630

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		snex 4613	. . 3 |- { A } e. _V
2		eleq2 2518	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
3		intid.1	. . . . 5 |- A e. _V
4	3	snid 3963	. . . 4 |- A e. { A }
5	2, 4	intmin3 4232	. . 3 |- ( { A } e. _V -> |^| { x | A e. x } C_ { A } )
6	1, 5	ax-mp 5	. 2 |- |^| { x | A e. x } C_ { A }
7	3	elintab 4214	. . . 4 |- ( A e. |^| { x | A e. x } <-> A. x ( A e. x -> A e. x ) )
8		id 22	. . . 4 |- ( A e. x -> A e. x )
9	7, 8	mpgbir 1676	. . 3 |- A e. |^| { x | A e. x }
10		snssi 4084	. . 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 3415	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1447   e. wcel 1890   {cab 2437   _Vcvv 3012   C_ wss 3371   {csn 3935   |^|cint 4203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-v 3014  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-sn 3936  df-pr 3938  df-int 4204

This theorem is referenced by: (None)