Theorem exel

Description: There exist two sets, one a member of the other.

This theorem looks similar to el , but its meaning is different. It only depends on the axioms ax-mp to ax-4 , ax-6 , and ax-pr . This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq . (Contributed by SN, 23-Dec-2024)

	Ref	Expression
Assertion	exel	$$⊢\exists y\exists xx\in y$$

Proof

Step	Hyp	Ref	Expression
1		ax-pr	$$⊢\exists y\forall x\left(\left(x=z\vee x=z\right)\to x\in y\right)$$
2		ax6ev	$$⊢\exists xx=z$$
3		pm2.07	$$⊢x=z\to \left(x=z\vee x=z\right)$$
4	2 3	eximii	$$⊢\exists x\left(x=z\vee x=z\right)$$
5		exim	$$⊢\forall x\left(\left(x=z\vee x=z\right)\to x\in y\right)\to \left(\exists x\left(x=z\vee x=z\right)\to \exists xx\in y\right)$$
6	4 5	mpi	$$⊢\forall x\left(\left(x=z\vee x=z\right)\to x\in y\right)\to \exists xx\in y$$
7	1 6	eximii	$$⊢\exists y\exists xx\in y$$