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	$$⊢\exists yx\in y$$

Proof

Step	Hyp	Ref	Expression
1		ax-pr	$$⊢\exists y\forall z\left(\left(z=x\vee z=x\right)\to z\in y\right)$$
2		orc	$$⊢z=x\to \left(z=x\vee z=x\right)$$
3		ax8v1	$$⊢z=x\to \left(z\in y\to x\in y\right)$$
4	2 3	embantd	$$⊢z=x\to \left(\left(\left(z=x\vee z=x\right)\to z\in y\right)\to x\in y\right)$$
5	4	spimvw	$$⊢\forall z\left(\left(z=x\vee z=x\right)\to z\in y\right)\to x\in y$$
6	1 5	eximii	$$⊢\exists yx\in y$$