Theorem 0ellim 4543

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

Assertion

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

Proof of Theorem 0ellim

Step	Hyp	Ref	Expression
1		dflim2 4515	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
2	1	simp2bi 1044	1 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ∅c0 3520  ∪ cuni 3935  Ord word 4507  Lim wlim 4509

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011  df-ilim 4514

This theorem is used by: (None)