Theorem dfss 3011

Description: Variant of subclass definition df-ss 3010. (Contributed by NM, 3-Sep-2004.)

Assertion

Ref	Expression
dfss	|- ( A C_ B <-> A = ( A i^i B ) )

Proof of Theorem dfss

Step	Hyp	Ref	Expression
1		df-ss 3010	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		eqcom 2090	. 2 |- ( ( A i^i B ) = A <-> A = ( A i^i B ) )
3	1, 2	bitri 182	1 |- ( A C_ B <-> A = ( A i^i B ) )

Syntax hints:   <-> wb 103   = wceq 1289   i^i cin 2996   C_ wss 2997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-ss 3010

This theorem is referenced by:  dfss2  3012  onelini  4248  cnvcnv  4870  funimass1  5077  sbthlemi5  6649  dmaddpi  6863  dmmulpi  6864