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Theorem undisj2 3680
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 3663 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
2 indi 3587 . . 3  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
32eqeq1i 2443 . 2  |-  ( ( A  i^i  ( B  u.  C ) )  =  (/)  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
41, 3bitr4i 244 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    u. cun 3318    i^i cin 3319   (/)c0 3628
This theorem is referenced by:  f1oun2prg  11864
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629
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