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Theorem undisj1 3671
Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
undisj1  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
C )  =  (/) )

Proof of Theorem undisj1
StepHypRef Expression
1 un00 3655 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
2 indir 3581 . . 3  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
32eqeq1i 2442 . 2  |-  ( ( ( A  u.  B
)  i^i  C )  =  (/)  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
41, 3bitr4i 244 1  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    u. cun 3310    i^i cin 3311   (/)c0 3620
This theorem is referenced by:  funtp  5495  f1oun2prg  11856
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-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621
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