Theorem dfss 3136

Description: Variant of subclass definition df-ss 3135. (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 3135	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		eqcom 2173	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ 𝐴 = (𝐴 ∩ 𝐵))
3	1, 2	bitri 183	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))

Syntax hints:   ↔ wb 104   = wceq 1349   ∩ cin 3121   ⊆ wss 3122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-gen 1443  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-cleq 2164  df-ss 3135

This theorem is referenced by:  dfss2  3137  onelini  4416  cnvcnv  5065  funimass1  5277  sbthlemi5  6942  dmaddpi  7291  dmmulpi  7292  tgioo  13425