Theorem ssequn2 3295

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 3292	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3266	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2173	. 2 |- ( ( A u. B ) = B <-> ( B u. A ) = B )
4	1, 3	bitri 183	1 |- ( A C_ B <-> ( B u. A ) = B )

Syntax hints:   <-> wb 104   = wceq 1343   u. cun 3114   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by:  unabs  3353  pwssunim  4262  pwundifss  4263  oneluni  4409  relresfld  5133  relcoi1  5135  fsnunf  5685  unsnfidcel  6886  fidcenumlemr  6920  exmidfodomrlemim  7157  ennnfonelemhf1o  12346