Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1138 Unicode version

Theorem bnj1138 29136
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1138.1  |-  A  =  ( B  u.  C
)
Assertion
Ref Expression
bnj1138  |-  ( X  e.  A  <->  ( X  e.  B  \/  X  e.  C ) )

Proof of Theorem bnj1138
StepHypRef Expression
1 bnj1138.1 . . 3  |-  A  =  ( B  u.  C
)
21eleq2i 2360 . 2  |-  ( X  e.  A  <->  X  e.  ( B  u.  C
) )
3 elun 3329 . 2  |-  ( X  e.  ( B  u.  C )  <->  ( X  e.  B  \/  X  e.  C ) )
42, 3bitri 240 1  |-  ( X  e.  A  <->  ( X  e.  B  \/  X  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696    u. cun 3163
This theorem is referenced by:  bnj1424  29187  bnj1408  29382  bnj1417  29387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
  Copyright terms: Public domain W3C validator