Theorem difun1 3415

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 3365	. . . 4 |- ( ( A i^i ( _V \ B ) ) i^i ( _V \ C ) ) = ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) )
2		invdif 3397	. . . 4 |- ( ( A i^i ( _V \ B ) ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ B ) ) \ C )
3	1, 2	eqtr3i 2212	. . 3 |- ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) ) = ( ( A i^i ( _V \ B ) ) \ C )
4		undm 3413	. . . . 5 |- ( _V \ ( B u. C ) ) = ( ( _V \ B ) i^i ( _V \ C ) )
5	4	ineq2i 3353	. . . 4 |- ( A i^i ( _V \ ( B u. C ) ) ) = ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) )
6		invdif 3397	. . . 4 |- ( A i^i ( _V \ ( B u. C ) ) ) = ( A \ ( B u. C ) )
7	5, 6	eqtr3i 2212	. . 3 |- ( A i^i ( ( _V \ B ) i^i ( _V \ C ) ) ) = ( A \ ( B u. C ) )
8	3, 7	eqtr3i 2212	. 2 |- ( ( A i^i ( _V \ B ) ) \ C ) = ( A \ ( B u. C ) )
9		invdif 3397	. . 3 |- ( A i^i ( _V \ B ) ) = ( A \ B )
10	9	difeq1i 3269	. 2 |- ( ( A i^i ( _V \ B ) ) \ C ) = ( ( A \ B ) \ C )
11	8, 10	eqtr3i 2212	1 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )

Syntax hints:   = wceq 1364   _Vcvv 2756   \ cdif 3146   u. cun 3147   i^i cin 3148

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-rab 2477  df-v 2758  df-dif 3151  df-un 3153  df-in 3155

This theorem is referenced by:  dif32  3418  difabs  3419  difpr  3756  diffifi  6937  difinfinf  7146