Theorem dfss1 3285

Description: A frequently-used variant of subclass definition df-ss 3089. (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 3089	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3273	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2148	. 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 1332   i^i cin 3075   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by:  dfss5  3286  sseqin2  3300  onintexmid  4495  xpimasn  4995  fndmdif  5533  infiexmid  6779  ssfidc  6831  isumss  11192  znnen  11947