Theorem ssequn2 3145

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 3142	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3116	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2088	. 2 |- ( ( A u. B ) = B <-> ( B u. A ) = B )
4	1, 3	bitri 182	1 |- ( A C_ B <-> ( B u. A ) = B )

Syntax hints:   <-> wb 103   = wceq 1284   u. cun 2971   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by:  unabs  3196  pwssunim  4039  pwundifss  4040  oneluni  4186  relresfld  4867  relcoi1  4869  fsnunf  5383  unsnfidcel  6386