Theorem dfss 3130

Description: Variant of subclass definition df-ss 3129. (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 3129	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		eqcom 2167	. 2 |- ( ( A i^i B ) = A <-> A = ( A i^i B ) )
3	1, 2	bitri 183	1 |- ( A C_ B <-> A = ( A i^i B ) )

Syntax hints:   <-> wb 104   = wceq 1343   i^i cin 3115   C_ wss 3116

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 1435  ax-gen 1437  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ss 3129

This theorem is referenced by:  dfss2  3131  onelini  4408  cnvcnv  5056  funimass1  5265  sbthlemi5  6926  dmaddpi  7266  dmmulpi  7267  tgioo  13186