Theorem fi0

Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015)

	Ref	Expression
Assertion	fi0	|- ( fi ` (/) ) = (/)

Proof

Step	Hyp	Ref	Expression
1		0ex	|- (/) e. _V
2		fival	|- ( (/) e. _V -> ( fi ` (/) ) = { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } )
3	1 2	ax-mp	|- ( fi ` (/) ) = { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x }
4		vprc	|- -. _V e. _V
5		id	|- ( y = |^| x -> y = |^| x )
6		elinel1	|- ( x e. ( ~P (/) i^i Fin ) -> x e. ~P (/) )
7		elpwi	|- ( x e. ~P (/) -> x C_ (/) )
8		ss0	|- ( x C_ (/) -> x = (/) )
9	6 7 8	3syl	|- ( x e. ( ~P (/) i^i Fin ) -> x = (/) )
10	9	inteqd	|- ( x e. ( ~P (/) i^i Fin ) -> |^| x = |^| (/) )
11		int0	|- |^| (/) = _V
12	10 11	eqtrdi	|- ( x e. ( ~P (/) i^i Fin ) -> |^| x = _V )
13	5 12	sylan9eqr	|- ( ( x e. ( ~P (/) i^i Fin ) /\ y = |^| x ) -> y = _V )
14	13	rexlimiva	|- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> y = _V )
15		vex	|- y e. _V
16	14 15	eqeltrrdi	|- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> _V e. _V )
17	4 16	mto	|- -. E. x e. ( ~P (/) i^i Fin ) y = |^| x
18	17	abf	|- { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } = (/)
19	3 18	eqtri	|- ( fi ` (/) ) = (/)