Theorem ssequn2 3380

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 3377	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3351	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2239	. 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 1397   u. cun 3198   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  unabs  3438  pwssunim  4381  pwundifss  4382  oneluni  4528  relresfld  5266  relcoi1  5268  fsnunf  5853  unsnfidcel  7112  tpfidceq  7121  fidcenumlemr  7153  exmidfodomrlemim  7411  ennnfonelemhf1o  13033  lspun0  14438  plyrecj  15486  dvply2g  15489