Theorem ssequn2 3529

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 3526	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3500	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2450	. 2 |- ( ( A u. B ) = B <-> ( B u. A ) = B )
4	1, 3	bitri 249	1 |- ( A C_ B <-> ( B u. A ) = B )

Syntax hints:   <-> wb 184   = wceq 1369   u. cun 3326   C_ wss 3328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-un 3333  df-in 3335  df-ss 3342

This theorem is referenced by:  unabs  3580  tppreqb  4014  pwssun  4627  pwundif  4628  ordssun  4818  ordequn  4819  oneluni  4831  relresfld  5364  relcoi1  5366  fsnunf  5916  sorpssun  6367  ordunpr  6437  fodomr  7462  enp1ilem  7546  pwfilem  7605  brwdom2  7788  sucprcreg  7814  dfacfin7  8568  hashbclem  12205  incexclem  13299  ramub1lem1  14087  ramub1lem2  14088  mreexmrid  14581  lspun0  17092  lbsextlem4  17242  cldlp  18754  ordtuni  18794  cldsubg  19681  trust  19804  nulmbl2  21018  limcmpt2  21359  cnplimc  21362  dvreslem  21384  dvaddbr  21412  dvmulbr  21413  lhop  21488  plypf1  21680  coeeulem  21692  coeeu  21693  coef2  21699  rlimcnp  22359  ex-un  23631  shs0i  24852  chj0i  24858  ffsrn  26029  difioo  26072  eulerpartlemt  26754  subfacp1lem1  27067  cvmscld  27162  refssfne  28566  topjoin  28586  istopclsd  29036  nacsfix  29048  diophrw  29097  stoweidlem44  29839