Theorem ssequn2 3402

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 3399	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3373	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2246	. 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 1402   u. cun 3218   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by:  unabs  3462  pwssunim  4424  pwundifss  4425  oneluni  4571  relresfld  5312  relcoi1  5314  fsnunf  5906  unsnfidcel  7218  tpfidceq  7227  fidcenumlemr  7262  exmidfodomrlemim  7543  ennnfonelemhf1o  13282  lspun0  14734  plyrecj  15787  dvply2g  15790