Theorem nlim1NEW

Description: 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024) (Proof shortened by RP, 13-Dec-2024)

	Ref	Expression
Assertion	nlim1NEW	⊢ ¬ Lim 1o

Proof

Step	Hyp	Ref	Expression
1		0elon	⊢ ∅ ∈ On
2		nlimsuc	⊢ ( ∅ ∈ On → ¬ Lim suc ∅ )
3		df-1o	⊢ 1o = suc ∅
4		limeq	⊢ ( 1o = suc ∅ → ( Lim 1o ↔ Lim suc ∅ ) )
5	3 4	ax-mp	⊢ ( Lim 1o ↔ Lim suc ∅ )
6	2 5	sylnibr	⊢ ( ∅ ∈ On → ¬ Lim 1o )
7	1 6	ax-mp	⊢ ¬ Lim 1o