Theorem ssequn2 3333

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 3330	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3304	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2201	. 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 1364   u. cun 3152   C_ wss 3154

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167

This theorem is referenced by:  unabs  3391  pwssunim  4316  pwundifss  4317  oneluni  4463  relresfld  5196  relcoi1  5198  fsnunf  5759  unsnfidcel  6979  fidcenumlemr  7016  exmidfodomrlemim  7263  ennnfonelemhf1o  12573  lspun0  13924  plyrecj  14941