Theorem ssequn2 3377

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 3374	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3348	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2237	. 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 1395   u. cun 3195   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-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  unabs  3435  pwssunim  4374  pwundifss  4375  oneluni  4521  relresfld  5257  relcoi1  5259  fsnunf  5838  unsnfidcel  7079  tpfidceq  7088  fidcenumlemr  7118  exmidfodomrlemim  7375  ennnfonelemhf1o  12979  lspun0  14383  plyrecj  15431  dvply2g  15434