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	|- E. y E. x x e. y

Proof

Step	Hyp	Ref	Expression
1		ax-pr	|- E. y A. x ( ( x = z \/ x = z ) -> x e. y )
2		ax6ev	|- E. x x = z
3		pm2.07	|- ( x = z -> ( x = z \/ x = z ) )
4	2 3	eximii	|- E. x ( x = z \/ x = z )
5		exim	|- ( A. x ( ( x = z \/ x = z ) -> x e. y ) -> ( E. x ( x = z \/ x = z ) -> E. x x e. y ) )
6	4 5	mpi	|- ( A. x ( ( x = z \/ x = z ) -> x e. y ) -> E. x x e. y )
7	1 6	eximii	|- E. y E. x x e. y