Theorem intid 4258

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 4219	. . 3 |- { A } e. _V
3		eleq2 2260	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
4	1	snid 3654	. . . 4 |- A e. { A }
5	3, 4	intmin3 3902	. . 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 3886	. . . 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 1467	. . 3 |- A e. |^| { x | A e. x }
10		snssi 3767	. . 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 3200	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   C_ wss 3157   {csn 3623   |^|cint 3875

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-int 3876

This theorem is referenced by: (None)