Theorem ssequn2 3345

Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Assertion

Ref	Expression
ssequn2	|- ( A C_ B <-> ( B u. A ) = B )

Proof of Theorem ssequn2

Step	Hyp	Ref	Expression
1		ssequn1 3342	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3316	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2212	. 2 |- ( ( A u. B ) = B <-> ( B u. A ) = B )
4	1, 3	bitri 184	1 |- ( A C_ B <-> ( B u. A ) = B )

Syntax hints:   <-> wb 105   = wceq 1372   u. cun 3163   C_ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by:  unabs  3403  pwssunim  4330  pwundifss  4331  oneluni  4477  relresfld  5211  relcoi1  5213  fsnunf  5783  unsnfidcel  7017  tpfidceq  7026  fidcenumlemr  7056  exmidfodomrlemim  7308  ennnfonelemhf1o  12755  lspun0  14158  plyrecj  15206  dvply2g  15209