Theorem difun1 3464

Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Assertion

Ref	Expression
difun1	|- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )

Proof of Theorem difun1

Step	Hyp	Ref	Expression
1		inass 3414	. . . 4 |- ( ( A i^i ( _V \ B ) ) i^i ( _V \ C ) ) = ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) )
2		invdif 3446	. . . 4 |- ( ( A i^i ( _V \ B ) ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ B ) ) \ C )
3	1, 2	eqtr3i 2252	. . 3 |- ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) ) = ( ( A i^i ( _V \ B ) ) \ C )
4		undm 3462	. . . . 5 |- ( _V \ ( B u. C ) ) = ( ( _V \ B ) i^i ( _V \ C ) )
5	4	ineq2i 3402	. . . 4 |- ( A i^i ( _V \ ( B u. C ) ) ) = ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) )
6		invdif 3446	. . . 4 |- ( A i^i ( _V \ ( B u. C ) ) ) = ( A \ ( B u. C ) )
7	5, 6	eqtr3i 2252	. . 3 |- ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) ) = ( A \ ( B u. C ) )
8	3, 7	eqtr3i 2252	. 2 |- ( ( A i^i ( _V \ B ) ) \ C ) = ( A \ ( B u. C ) )
9		invdif 3446	. . 3 |- ( A i^i ( _V \ B ) ) = ( A \ B )
10	9	difeq1i 3318	. 2 |- ( ( A i^i ( _V \ B ) ) \ C ) = ( ( A \ B ) \ C )
11	8, 10	eqtr3i 2252	1 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )

Syntax hints:   = wceq 1395   _Vcvv 2799   \ cdif 3194   u. cun 3195   i^i cin 3196

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203

This theorem is referenced by:  dif32  3467  difabs  3468  difpr  3810  diffifi  7056  difinfinf  7268