Theorem ssequn2 3346

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 3343	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3317	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2213	. 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 3164   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by:  unabs  3404  pwssunim  4331  pwundifss  4332  oneluni  4478  relresfld  5212  relcoi1  5214  fsnunf  5784  unsnfidcel  7018  tpfidceq  7027  fidcenumlemr  7057  exmidfodomrlemim  7309  ennnfonelemhf1o  12784  lspun0  14187  plyrecj  15235  dvply2g  15238