Theorem dfss1 3311

Description: A frequently-used variant of subclass definition df-ss 3115. (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 3115	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3299	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2165	. 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 1335   i^i cin 3101   C_ wss 3102

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115

This theorem is referenced by:  dfss5  3312  sseqin2  3326  onintexmid  4533  xpimasn  5035  fndmdif  5573  infiexmid  6823  ssfidc  6880  isumss  11292  znnen  12169