Theorem intid 4695

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 4678	. . 3 |- { A } e. _V
2		eleq2 2527	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
3		intid.1	. . . . 5 |- A e. _V
4	3	snid 4044	. . . 4 |- A e. { A }
5	2, 4	intmin3 4300	. . 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 4282	. . . 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 1627	. . 3 |- A e. |^| { x | A e. x }
10		snssi 4160	. . 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 3505	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   {cab 2439   _Vcvv 3106   C_ wss 3461   {csn 4016   |^|cint 4271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-int 4272

This theorem is referenced by: (None)