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Theorem undisj1 3519
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 3503 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
2 indir 3430 . . 3  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
32eqeq1i 2303 . 2  |-  ( ( ( A  u.  B
)  i^i  C )  =  (/)  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
41, 3bitr4i 243 1  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    u. cun 3163    i^i cin 3164   (/)c0 3468
This theorem is referenced by:  funtp  5319  f1oun2prg  28187
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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469
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