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Theorem undisj1 3388
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 3377 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  i^i  C )  u.  ( B  i^i  C
) )  =  (/) )
2 indir 3293 . . 3  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
32eqeq1i 2123 . 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 1314    u. cun 3037    i^i cin 3038   (/)c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by:  funtp  5144
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