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Metamath Proof Explorer

Definition df-uni

Description: Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of TakeutiZaring p. 16. For example, U. { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } ( ex-uni ). This is similar to the union of two classes df-un . (Contributed by NM, 23-Aug-1993)

		Ref	Expression
	Assertion	df-uni	\|- U. A = { x \| E. y ( x e. y /\ y e. A ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	\|- A
1	0	cuni	\|- U. A
2		vx	\|- x
3		vy	\|- y
4	2	cv	\|- x
5	3	cv	\|- y
6	4 5	wcel	\|- x e. y
7	5 0	wcel	\|- y e. A
8	6 7	wa	\|- ( x e. y /\ y e. A )
9	8 3	wex	\|- E. y ( x e. y /\ y e. A )
10	9 2	cab	\|- { x \| E. y ( x e. y /\ y e. A ) }
11	1 10	wceq	\|- U. A = { x \| E. y ( x e. y /\ y e. A ) }