Theorem nlim0 2282

Description: The empty set is not a limit ordinal.

Assertion

Ref	Expression
nlim0	⊢ ¬ Lim ∅

Proof of Theorem nlim0

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 ⊢ ∅ = ∅
2		df-lim 2204	. . 3 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ¬ ∅ = ∅ ∧ ∅ = ∪∅))
3		3simp2 595	. . 3 ⊢ ((Ord ∅ ∧ ¬ ∅ = ∅ ∧ ∅ = ∪∅) → ¬ ∅ = ∅)
4	2, 3	sylbi 174	. 2 ⊢ (Lim ∅ → ¬ ∅ = ∅)
5	1, 4	mt2 96	1 ⊢ ¬ Lim ∅

Syntax hints:  ¬ wn 1   ∧ w3a 581   = wceq 1091  ∅c0 1707  ∪cuni 1919  Ord word 2198  Lim wlim 2200

This theorem is referenced by:  0ellim 2285  dflim3 2368  tz7.44lem1 2965

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-gen 677  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-cleq 1097  df-lim 2204

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