Theorem ssequn2 3213

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 3210	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3184	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2120	. 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 1312   u. cun 3033   C_ wss 3035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048

This theorem is referenced by:  unabs  3271  pwssunim  4164  pwundifss  4165  oneluni  4311  relresfld  5024  relcoi1  5026  fsnunf  5572  unsnfidcel  6759  fidcenumlemr  6792  exmidfodomrlemim  7001  ennnfonelemhf1o  11764