Theorem dfss1 3332

Description: A frequently-used variant of subclass definition df-ss 3135. (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 3135	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		incom 3320	. . 3 |- ( A i^i B ) = ( B i^i A )
3	2	eqeq1i 2179	. 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 1349   i^i cin 3121   C_ wss 3122

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-in 3128  df-ss 3135

This theorem is referenced by:  dfss5  3333  sseqin2  3347  onintexmid  4558  xpimasn  5061  fndmdif  5605  infiexmid  6859  ssfidc  6916  isumss  11358  znnen  12357