Theorem risset

Description: Two ways to say " A belongs to B ". (Contributed by NM, 22-Nov-1994)

	Ref	Expression
Assertion	risset	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		exancom	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
2		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) )
3		dfclel	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
4	1 2 3	3bitr4ri	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 )