Theorem ssequn2 3159

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 3156	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3130	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2092	. 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 1287   u. cun 2984   C_ wss 2986

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990  df-in 2992  df-ss 2999

This theorem is referenced by:  unabs  3216  pwssunim  4078  pwundifss  4079  oneluni  4225  relresfld  4917  relcoi1  4919  fsnunf  5441  unsnfidcel  6561  exmidfodomrlemim  6748