MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssequn2 Structured version   Unicode version

Theorem ssequn2 3512
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
StepHypRef Expression
1 ssequn1 3509 . 2  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
2 uncom 3483 . . 3  |-  ( A  u.  B )  =  ( B  u.  A
)
32eqeq1i 2442 . 2  |-  ( ( A  u.  B )  =  B  <->  ( B  u.  A )  =  B )
41, 3bitri 241 1  |-  ( A 
C_  B  <->  ( B  u.  A )  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    u. cun 3310    C_ wss 3312
This theorem is referenced by:  unabs  3563  tppreqb  3931  pwssun  4481  pwundif  4482  ordssun  4673  ordequn  4674  oneluni  4686  ordunpr  4798  relresfld  5388  relcoi1  5390  fsnunf  5923  sorpssun  6521  fodomr  7250  enp1ilem  7334  pwfilem  7393  brwdom2  7533  dfacfin7  8271  hashbclem  11693  incexclem  12608  ramub1lem1  13386  ramub1lem2  13387  mreexmrid  13860  lspun0  16079  lbsextlem4  16225  cldlp  17206  ordtuni  17246  cldsubg  18132  trust  18251  nulmbl2  19423  limcmpt2  19763  cnplimc  19766  dvreslem  19788  dvaddbr  19816  dvmulbr  19817  lhop  19892  plypf1  20123  coeeulem  20135  coeeu  20136  coef2  20142  rlimcnp  20796  ex-un  21724  shs0i  22943  chj0i  22949  difioo  24137  subfacp1lem1  24857  cvmscld  24952  refssfne  26365  topjoin  26385  istopclsd  26745  nacsfix  26757  diophrw  26808  stoweidlem44  27760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326
  Copyright terms: Public domain W3C validator