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 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 𝑋 = 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		risset	⊢ ( 𝑋 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 𝑥 = 𝑋 )
2		eqcom	⊢ ( 𝑥 = 𝑋 ↔ 𝑋 = 𝑥 )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃ 𝑥 ∈ 𝐴 𝑋 = 𝑥 )
4	1 3	bitri	⊢ ( 𝑋 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 𝑋 = 𝑥 )