Theorem onelini

Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994)

	Ref	Expression
Hypothesis	on.1	⊢ 𝐴 ∈ On
Assertion	onelini	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 = ( 𝐵 ∩ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2	1	onelssi	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 )
3		dfss	⊢ ( 𝐵 ⊆ 𝐴 ↔ 𝐵 = ( 𝐵 ∩ 𝐴 ) )
4	2 3	sylib	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 = ( 𝐵 ∩ 𝐴 ) )