Theorem onssnel2i

Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994)

	Ref	Expression
Hypothesis	on.1	⊢ 𝐴 ∈ On
Assertion	onssnel2i	⊢ ( 𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2	1	onirri	⊢ ¬ 𝐴 ∈ 𝐴
3		ssel	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐴 ) )
4	2 3	mtoi	⊢ ( 𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵 )