Theorem el

Description: Any 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) Use ax-pr instead of ax-9 and ax-pow . (Revised by BTernaryTau, 2-Dec-2024) (Proof shortened by Matthew House, 6-Apr-2026)

	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		orc	|- ( z = x -> ( z = x \/ z = x ) )
3		ax8v1	|- ( z = x -> ( z e. y -> x e. y ) )
4	2 3	embantd	|- ( z = x -> ( ( ( z = x \/ z = x ) -> z e. y ) -> x e. y ) )
5	4	spimvw	|- ( A. z ( ( z = x \/ z = x ) -> z e. y ) -> x e. y )
6	1 5	eximii	|- E. y x e. y