Theorem dfss 3261

Description: Variant of subclass definition df-ss 3260. (Contributed by NM, 3-Sep-2004.)

Assertion

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

Proof of Theorem dfss

Step	Hyp	Ref	Expression
1		df-ss 3260	. 2 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
2		eqcom 2355	. 2 ⊢ ((A ∩ B) = A ↔ A = (A ∩ B))
3	1, 2	bitri 240	1 ⊢ (A ⊆ B ↔ A = (A ∩ B))

This proof depends on syntax axioms:   ↔ wb 176   = wceq 1642   ∩ cin 3209   ⊆ wss 3258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-cleq 2346  df-ss 3260

This theorem is used by:  dfss2  3263  iinrab2  4030  funimass1  5170