Theorem elex

Description: If a class is a member of another class, then it is a set. Theorem 6.12 of Quine p. 44. (Contributed by NM, 26-May-1993) (Proof shortened by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Assertion	elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		exsimpl	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑥 𝑥 = 𝐴 )
2		dfclel	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
3		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
4	1 2 3	3imtr4i	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )