MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uneq1 Structured version   Unicode version

Theorem uneq1 3486
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )

Proof of Theorem uneq1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2496 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21orbi1d 684 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  \/  x  e.  C
)  <->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3480 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3480 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43bitr4g 280 . 2  |-  ( A  =  B  ->  (
x  e.  ( A  u.  C )  <->  x  e.  ( B  u.  C
) ) )
65eqrdv 2433 1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725    u. cun 3310
This theorem is referenced by:  uneq2  3487  uneq12  3488  uneq1i  3489  uneq1d  3492  unineq  3583  prprc1  3906  uniprg  4022  unexb  4701  relresfld  5388  relcoi1  5390  oarec  6797  xpider  6967  undifixp  7090  unxpdom  7308  enp1ilem  7334  findcard2  7340  domunfican  7371  pwfilem  7393  fin1a2lem10  8281  incexclem  12608  ramub1lem1  13386  ramub1  13388  mreexexlem3d  13863  mreexexlem4d  13864  ipodrsima  14583  mplsubglem  16490  mretopd  17148  iscldtop  17151  nconsubb  17478  plyval  20104  spanun  23039  difeq  23990  measun  24557  nofulllem2  25650  brsuccf  25778  altopthsn  25798  rankung  26099  ralxpmap  26733  nacsfix  26757  eldioph4b  26863  eldioph4i  26864  diophren  26865  compne  27610  islshp  29714  lshpset2N  29854  paddval  30532
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
  Copyright terms: Public domain W3C validator