Theorem 0ellim 4400

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 4372	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
2	1	simp2bi 1013	1 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  ∅c0 3424  ∪ cuni 3811  Ord word 4364  Lim wlim 4366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-3an 980  df-ilim 4371

This theorem is referenced by: (None)