Theorem onsseli

Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995)

	Ref	Expression
Hypotheses	on.1	⊢ 𝐴 ∈ On
	on.2	⊢ 𝐵 ∈ On
Assertion	onsseli	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2		on.2	⊢ 𝐵 ∈ On
3		onsseleq	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) )