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

Theorem uneqri 3292
Description: Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
uneqri.1  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )
Assertion
Ref Expression
uneqri  |-  ( A  u.  B )  =  C
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem uneqri
StepHypRef Expression
1 elun 3291 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
2 uneqri.1 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )
31, 2bitri 242 . 2  |-  ( x  e.  ( A  u.  B )  <->  x  e.  C )
43eqriv 2255 1  |-  ( A  u.  B )  =  C
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621    u. cun 3125
This theorem is referenced by:  unidm  3293  uncom  3294  unass  3307  dfun2  3379  undi  3391  unab  3410  un0  3454  inundif  3507
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-un 3132
  Copyright terms: Public domain W3C validator