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Theorem undisj2 3507
Description: The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
undisj2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )

Proof of Theorem undisj2
StepHypRef Expression
1 un00 3490 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
2 indi 3415 . . 3  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
32eqeq1i 2290 . 2  |-  ( ( A  i^i  ( B  u.  C ) )  =  (/)  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
41, 3bitr4i 243 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    u. cun 3150    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  f1oun2prg  28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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