Theorem ssequn2 3670

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 3667	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3641	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2467	. 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 1374   u. cun 3467   C_ wss 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-in 3476  df-ss 3483

This theorem is referenced by:  unabs  3721  tppreqb  4161  pwssun  4779  pwundif  4780  ordssun  4970  ordequn  4971  oneluni  4983  relresfld  5525  relcoi1  5527  fsnunf  6090  sorpssun  6562  ordunpr  6632  fodomr  7658  enp1ilem  7743  pwfilem  7803  brwdom2  7988  sucprcreg  8014  dfacfin7  8768  hashbclem  12454  incexclem  13600  ramub1lem1  14392  ramub1lem2  14393  mreexmrid  14887  lspun0  17433  lbsextlem4  17583  cldlp  19410  ordtuni  19450  cldsubg  20337  trust  20460  nulmbl2  21675  limcmpt2  22016  cnplimc  22019  dvreslem  22041  dvaddbr  22069  dvmulbr  22070  lhop  22145  plypf1  22337  coeeulem  22349  coeeu  22350  coef2  22356  rlimcnp  23016  ex-un  24808  shs0i  26029  chj0i  26035  ffsrn  27210  difioo  27247  eulerpartlemt  27936  subfacp1lem1  28249  cvmscld  28344  refssfne  29753  topjoin  29773  istopclsd  30223  nacsfix  30235  diophrw  30283  limciccioolb  31118  limcicciooub  31134  ioccncflimc  31179  icocncflimc  31183  cncfiooicclem1  31187  stoweidlem44  31299  dirkercncflem3  31360  fourierdlem62  31424