Theorem iotassuni

Description: The iota class is a subset of the union of all elements satisfying ph . (Contributed by Mario Carneiro, 24-Dec-2016) Remove dependency on ax-10 , ax-11 , ax-12 . (Revised by SN, 6-Nov-2024)

	Ref	Expression
Assertion	iotassuni	|- ( iota x ph ) C_ U. { x | ph }

Proof

Step	Hyp	Ref	Expression
1		iotauni2	|- ( E. y { x | ph } = { y } -> ( iota x ph ) = U. { x | ph } )
2		eqimss	|- ( ( iota x ph ) = U. { x | ph } -> ( iota x ph ) C_ U. { x | ph } )
3	1 2	syl	|- ( E. y { x | ph } = { y } -> ( iota x ph ) C_ U. { x | ph } )
4		iotanul2	|- ( -. E. y { x | ph } = { y } -> ( iota x ph ) = (/) )
5		0ss	|- (/) C_ U. { x | ph }
6	4 5	eqsstrdi	|- ( -. E. y { x | ph } = { y } -> ( iota x ph ) C_ U. { x | ph } )
7	3 6	pm2.61i	|- ( iota x ph ) C_ U. { x | ph }