Theorem intid 4926

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 4908	. . 3 |- { A } e. _V
2		eleq2 2690	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
3		intid.1	. . . . 5 |- A e. _V
4	3	snid 4208	. . . 4 |- A e. { A }
5	2, 4	intmin3 4505	. . 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 4487	. . . 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 1726	. . 3 |- A e. |^| { x | A e. x }
10		snssi 4339	. . 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 3619	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   {csn 4177   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-int 4476

This theorem is referenced by: (None)