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	$$⊢\exists yx\in y$$

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		pm4.25	$$⊢z=x\leftrightarrow \left(z=x\vee z=x\right)$$
3	2	imbi1i	$$⊢\left(z=x\to z\in y\right)\leftrightarrow \left(\left(z=x\vee z=x\right)\to z\in y\right)$$
4	3	albii	$$⊢\forall z\left(z=x\to z\in y\right)\leftrightarrow \forall z\left(\left(z=x\vee z=x\right)\to z\in y\right)$$
5		elequ1	$$⊢z=x\to \left(z\in y\leftrightarrow x\in y\right)$$
6	5	equsalvw	$$⊢\forall z\left(z=x\to z\in y\right)\leftrightarrow x\in y$$
7	4 6	bitr3i	$$⊢\forall z\left(\left(z=x\vee z=x\right)\to z\in y\right)\leftrightarrow x\in y$$
8	7	exbii	$$⊢\exists y\forall z\left(\left(z=x\vee z=x\right)\to z\in y\right)\leftrightarrow \exists yx\in y$$
9	1 8	mpbi	$$⊢\exists yx\in y$$