Theorem el

Description: Every set is an element of some other set. See elALT for a shorter proof using more axioms, and see elALT2 for a proof that uses ax-9 and ax-pow instead of ax-pr . (Contributed by NM, 4-Jan-2002) (Proof shortened by Andrew Salmon, 25-Jul-2011) Avoid ax-9 , ax-pow . (Revised by BTernaryTau, 2-Dec-2024)

	Ref	Expression
Assertion	el	|- E. y x e. y

Proof

Step	Hyp	Ref	Expression
1		ax-pr	|- E. y A. z ( ( z = x \/ z = x ) -> z e. y )
2		pm4.25	|- ( z = x <-> ( z = x \/ z = x ) )
3	2	imbi1i	|- ( ( z = x -> z e. y ) <-> ( ( z = x \/ z = x ) -> z e. y ) )
4	3	albii	|- ( A. z ( z = x -> z e. y ) <-> A. z ( ( z = x \/ z = x ) -> z e. y ) )
5		elequ1	|- ( z = x -> ( z e. y <-> x e. y ) )
6	5	equsalvw	|- ( A. z ( z = x -> z e. y ) <-> x e. y )
7	4 6	bitr3i	|- ( A. z ( ( z = x \/ z = x ) -> z e. y ) <-> x e. y )
8	7	exbii	|- ( E. y A. z ( ( z = x \/ z = x ) -> z e. y ) <-> E. y x e. y )
9	1 8	mpbi	|- E. y x e. y