Theorem dfss

Description: Variant of subclass definition df-ss . (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	dfss	⊢ ( 𝐴 ⊆ 𝐵 ↔ 𝐴 = ( 𝐴 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 )
2		eqcom	⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐴 ↔ 𝐴 = ( 𝐴 ∩ 𝐵 ) )
3	1 2	bitri	⊢ ( 𝐴 ⊆ 𝐵 ↔ 𝐴 = ( 𝐴 ∩ 𝐵 ) )