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Theorem undisj2 3496
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 3484 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
2 indi 3397 . . 3  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
32eqeq1i 2197 . 2  |-  ( ( A  i^i  ( B  u.  C ) )  =  (/)  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
41, 3bitr4i 187 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    u. cun 3142    i^i cin 3143   (/)c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438
This theorem is referenced by: (None)
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