Theorem ssequn2 3354

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 3351	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3325	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2215	. 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 1373   u. cun 3172   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187

This theorem is referenced by:  unabs  3412  pwssunim  4349  pwundifss  4350  oneluni  4496  relresfld  5231  relcoi1  5233  fsnunf  5807  unsnfidcel  7044  tpfidceq  7053  fidcenumlemr  7083  exmidfodomrlemim  7340  ennnfonelemhf1o  12899  lspun0  14302  plyrecj  15350  dvply2g  15353