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	$$⊢\neg \mathrm{Lim}{1}_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		0elon	$$⊢\varnothing \in \mathrm{On}$$
2		nlimsuc	$$⊢\varnothing \in \mathrm{On}\to \neg \mathrm{Lim}\mathrm{suc}\varnothing$$
3		df-1o	$$⊢1_{𝑜}=\mathrm{suc}\varnothing$$
4		limeq	$$⊢1_{𝑜}=\mathrm{suc}\varnothing \to \left(\mathrm{Lim}{1}_{𝑜}\leftrightarrow \mathrm{Lim}\mathrm{suc}\varnothing \right)$$
5	3 4	ax-mp	$$⊢\mathrm{Lim}{1}_{𝑜}\leftrightarrow \mathrm{Lim}\mathrm{suc}\varnothing$$
6	2 5	sylnibr	$$⊢\varnothing \in \mathrm{On}\to \neg \mathrm{Lim}{1}_{𝑜}$$
7	1 6	ax-mp	$$⊢\neg \mathrm{Lim}{1}_{𝑜}$$