Theorem dfss 3167

Description: Variant of subclass definition df-ss 3166. (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 3166	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		eqcom 2195	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3	1, 2	bitri 184	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))

Syntax hints:   ↔ wb 105   = wceq 1364   ∩ cin 3152   ⊆ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ss 3166

This theorem is referenced by:  dfss2  3168  onelini  4461  cnvcnv  5118  funimass1  5331  sbthlemi5  7020  dmaddpi  7385  dmmulpi  7386  tgioo  14714