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 ( 𝐴𝐵𝐴 ∈ V )

Proof

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