Theorem intid 4018

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 3987	. . 3 |- { A } e. _V
3		eleq2 2148	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
4	1	snid 3452	. . . 4 |- A e. { A }
5	3, 4	intmin3 3692	. . 3 |- ( { A } e. _V -> |^| { x | A e. x } C_ { A } )
6	2, 5	ax-mp 7	. 2 |- |^| { x | A e. x } C_ { A }
7	1	elintab 3676	. . . 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 1385	. . 3 |- A e. |^| { x | A e. x }
10		snssi 3558	. . 3 |- ( A e. |^| { x | A e. x } -> { A } C_ |^| { x | A e. x } )
11	9, 10	ax-mp 7	. 2 |- { A } C_ |^| { x | A e. x }
12	6, 11	eqssi 3028	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1287   e. wcel 1436   {cab 2071   _Vcvv 2614   C_ wss 2986   {csn 3425   |^|cint 3665

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-int 3666

This theorem is referenced by: (None)