Theorem intmin2

Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Hypothesis	intmin2.1	|- A e. _V
Assertion	intmin2	|- |^| { x | A C_ x } = A

Proof

Step	Hyp	Ref	Expression
1		intmin2.1	|- A e. _V
2		rabab	|- { x e. _V | A C_ x } = { x | A C_ x }
3	2	inteqi	|- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
4		intmin	|- ( A e. _V -> |^| { x e. _V | A C_ x } = A )
5	1 4	ax-mp	|- |^| { x e. _V | A C_ x } = A
6	3 5	eqtr3i	|- |^| { x | A C_ x } = A