Theorem onnev

Description: The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007) (Proof shortened by Mario Carneiro, 10-Jan-2013) (Proof shortened by Wolf Lammen, 27-May-2024)

	Ref	Expression
Assertion	onnev	$$⊢\mathrm{On}\ne \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		snsn0non	$$⊢\neg \left\{\left\{\varnothing \right\}\right\}\in \mathrm{On}$$
2		snex	$$⊢\left\{\left\{\varnothing \right\}\right\}\in \mathrm{V}$$
3		id	$$⊢\mathrm{On}=\mathrm{V}\to \mathrm{On}=\mathrm{V}$$
4	2 3	eleqtrrid	$$⊢\mathrm{On}=\mathrm{V}\to \left\{\left\{\varnothing \right\}\right\}\in \mathrm{On}$$
5	1 4	mto	$$⊢\neg \mathrm{On}=\mathrm{V}$$
6	5	neir	$$⊢\mathrm{On}\ne \mathrm{V}$$