Theorem ssequn2 3392

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 3389	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3363	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2240	. 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 1398   u. cun 3209   C_ wss 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224

This theorem is referenced by:  unabs  3452  pwssunim  4405  pwundifss  4406  oneluni  4552  relresfld  5292  relcoi1  5294  fsnunf  5884  unsnfidcel  7181  tpfidceq  7190  fidcenumlemr  7225  exmidfodomrlemim  7504  ennnfonelemhf1o  13164  lspun0  14573  plyrecj  15628  dvply2g  15631