Theorem 1one2o

Description: Ordinal one is not ordinal two. Analogous to 1ne2 . (Contributed by AV, 17-Oct-2023)

	Ref	Expression
Assertion	1one2o	$$⊢1_{𝑜}\ne 2_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		1onn	$$⊢1_{𝑜}\in \mathrm{\omega }$$
2		omsucne	$$⊢1_{𝑜}\in \mathrm{\omega }\to 1_{𝑜}\ne \mathrm{suc}{1}_{𝑜}$$
3	1 2	ax-mp	$$⊢1_{𝑜}\ne \mathrm{suc}{1}_{𝑜}$$
4		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
5	3 4	neeqtrri	$$⊢1_{𝑜}\ne 2_{𝑜}$$