Theorem dfss1 3385

Description: A frequently-used variant of subclass definition df-ss 3187. (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 3187	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3373	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2215	. 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 1373   i^i cin 3173   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187

This theorem is referenced by:  dfss5  3386  sseqin2  3400  onintexmid  4639  xpimasn  5150  fndmdif  5708  infiexmid  7000  ssfidc  7060  isumss  11817  znnen  12884  2omap  16132  pw1map  16134