Theorem dfss1 3427

Description: A frequently-used variant of subclass definition df-ss 3226. (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 3226	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3413	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2242	. 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 1398   i^i cin 3212   C_ wss 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226

This theorem is referenced by:  dfss5  3428  sseqin2  3442  onintexmid  4697  xpimasn  5213  fndmdif  5785  infiexmid  7136  ssfidc  7200  2omap  7271  isumss  12081  znnen  13166  pw1map  16786