Theorem ssequn2 3244

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 3241	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3215	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2145	. 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 1331   u. cun 3064   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by:  unabs  3302  pwssunim  4201  pwundifss  4202  oneluni  4348  relresfld  5063  relcoi1  5065  fsnunf  5613  unsnfidcel  6802  fidcenumlemr  6836  exmidfodomrlemim  7050  ennnfonelemhf1o  11915