Theorem dfss1 3354

Description: A frequently-used variant of subclass definition df-ss 3157. (Contributed by NM, 10-Jan-2015.)

Assertion

Ref	Expression
dfss1	|- ( A C_ B <-> ( B i^i A ) = A )

Proof of Theorem dfss1

Step	Hyp	Ref	Expression
1		df-ss 3157	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3342	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2197	. 2 |- ( ( A i^i B ) = A <-> ( B i^i A ) = A )
4	1, 3	bitri 184	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1364   i^i cin 3143   C_ wss 3144

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157

This theorem is referenced by:  dfss5  3355  sseqin2  3369  onintexmid  4587  xpimasn  5092  fndmdif  5637  infiexmid  6895  ssfidc  6952  isumss  11417  znnen  12417