Theorem dfss 3171

Description: Variant of subclass definition df-ss 3170. (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 3170	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		eqcom 2198	. 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 1364   i^i cin 3156   C_ wss 3157

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 1461  ax-gen 1463  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ss 3170

This theorem is referenced by:  dfss2  3172  onelini  4465  cnvcnv  5122  funimass1  5335  sbthlemi5  7025  dmaddpi  7390  dmmulpi  7391  tgioo  14766