Theorem dfss5 3342

Description: Another definition of subclasshood. Similar to df-ss 3144, dfss 3145, and dfss1 3341. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
dfss5	|- ( A C_ B <-> A = ( B i^i A ) )

Proof of Theorem dfss5

Step	Hyp	Ref	Expression
1		dfss1 3341	. 2 |- ( A C_ B <-> ( B i^i A ) = A )
2		eqcom 2179	. 2 |- ( ( B i^i A ) = A <-> A = ( B i^i A ) )
3	1, 2	bitri 184	1 |- ( A C_ B <-> A = ( B i^i A ) )

Syntax hints:   <-> wb 105   = wceq 1353   i^i cin 3130   C_ wss 3131

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144

This theorem is referenced by:  nninfdcex  11956  nnmindc  12037  nnminle  12038