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Theorem undisj1 3878
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 3862 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
2 indir 3746 . . 3  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
32eqeq1i 2474 . 2  |-  ( ( ( A  u.  B
)  i^i  C )  =  (/)  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
41, 3bitr4i 252 1  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    u. cun 3474    i^i cin 3475   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by:  funtp  5640  f1oun2prg  12831
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