Theorem intid 3960

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		snexgOLD 3935	. . . 4 |- ( A e. _V -> { A } e. _V )
3	1, 2	ax-mp 7	. . 3 |- { A } e. _V
4		eleq2 2101	. . . 4 |- ( x = { A } -> ( A e. x <-> A e. { A } ) )
5	1	snid 3402	. . . 4 |- A e. { A }
6	4, 5	intmin3 3642	. . 3 |- ( { A } e. _V -> |^| { x | A e. x } C_ { A } )
7	3, 6	ax-mp 7	. 2 |- |^| { x | A e. x } C_ { A }
8	1	elintab 3626	. . . 4 |- ( A e. |^| { x | A e. x } <-> A. x ( A e. x -> A e. x ) )
9		id 19	. . . 4 |- ( A e. x -> A e. x )
10	8, 9	mpgbir 1342	. . 3 |- A e. |^| { x | A e. x }
11		snssi 3508	. . 3 |- ( A e. |^| { x | A e. x } -> { A } C_ |^| { x | A e. x } )
12	10, 11	ax-mp 7	. 2 |- { A } C_ |^| { x | A e. x }
13	7, 12	eqssi 2961	1 |- |^| { x | A e. x } = { A }

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {cab 2026   _Vcvv 2557   C_ wss 2917   {csn 3375   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-int 3616

This theorem is referenced by: (None)