Theorem dfss 3188

Description: Variant of subclass definition df-ss 3187. (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 3187	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		eqcom 2209	. 2 |- ( ( A i^i B ) = A <-> A = ( A i^i B ) )
3	1, 2	bitri 184	1 |- ( A C_ B <-> A = ( A i^i B ) )

Syntax hints:   <-> wb 105   = wceq 1373   i^i cin 3173   C_ wss 3174

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 1471  ax-gen 1473  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-ss 3187

This theorem is referenced by:  ssalel  3189  onelini  4495  cnvcnv  5154  funimass1  5370  sbthlemi5  7089  dmaddpi  7473  dmmulpi  7474  tgioo  15141