Theorem clel5

Description: Alternate definition of class membership: a class X is an element of another class A iff there is an element of A equal to X . (Contributed by AV, 13-Nov-2020) Remove use of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 19-May-2023)

	Ref	Expression
Assertion	clel5	$$⊢X\in A\leftrightarrow \exists x\in AX=x$$

Proof

Step	Hyp	Ref	Expression
1		risset	$$⊢X\in A\leftrightarrow \exists x\in Ax=X$$
2		eqcom	$$⊢x=X\leftrightarrow X=x$$
3	2	rexbii	$$⊢\exists x\in Ax=X\leftrightarrow \exists x\in AX=x$$
4	1 3	bitri	$$⊢X\in A\leftrightarrow \exists x\in AX=x$$