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Metamath Proof Explorer

Theorem 0ellim

Description: A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994)

Ref Expression
Assertion 0ellim ( Lim 𝐴 → ∅ ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 dflim2 ⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) )
2 1 simp2bi ⊢ ( Lim 𝐴 → ∅ ∈ 𝐴 )