Theorem dfss 3234

Description: Variant of subclass definition df-ss 3233. (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 3233	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		eqcom 2240	. 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 ) )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   i^i cin 3219   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ss 3233

This theorem is used by:  ssalel  3235  onelini  4575  cnvcnv  5240  funimass1  5458  sbthlemi5  7278  dmaddpi  7692  dmmulpi  7693  hashfibc  11283  tgioo  15655