Theorem dfss1 3408

Description: A frequently-used variant of subclass definition df-ss 3210. (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 3210	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3396	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2237	. 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 1395   i^i cin 3196   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210

This theorem is referenced by:  dfss5  3409  sseqin2  3423  onintexmid  4665  xpimasn  5177  fndmdif  5740  infiexmid  7039  ssfidc  7099  isumss  11902  znnen  12969  2omap  16359  pw1map  16361