Theorem dfss1 3326

Description: A frequently-used variant of subclass definition df-ss 3129. (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 3129	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3314	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2173	. 2 |- ( ( A i^i B ) = A <-> ( B i^i A ) = A )
4	1, 3	bitri 183	1 |- ( A C_ B <-> ( B i^i A ) = A )

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by:  dfss5  3327  sseqin2  3341  onintexmid  4550  xpimasn  5052  fndmdif  5590  infiexmid  6843  ssfidc  6900  isumss  11332  znnen  12331