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Theorem undisj1 3451
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 3440 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
2 indir 3356 . . 3  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
32eqeq1i 2165 . 2  |-  ( ( ( A  u.  B
)  i^i  C )  =  (/)  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
41, 3bitr4i 186 1  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335    u. cun 3100    i^i cin 3101   (/)c0 3394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by:  funtp  5222
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