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Metamath Proof Explorer

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) (Proof shortened by Wolf Lammen, 28-May-2025)

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

Proof

Step	Hyp	Ref	Expression
1		elissetv	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
2		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
3	1 2	sylibr	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )