Theorem nlim4

Description: 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024)

	Ref	Expression
Assertion	nlim4	$$⊢\neg \mathrm{Lim}{4}_{𝑜}$$

Proof

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