Theorem nlim2NEW

Description: 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024) (Proof shortened by RP, 13-Dec-2024)

	Ref	Expression
Assertion	nlim2NEW	$$⊢\neg \mathrm{Lim}{2}_{𝑜}$$

Proof

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