Theorem difun2 3504

Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
difun2	|- ( ( A u. B ) \ B ) = ( A \ B )

Proof of Theorem difun2

Step	Hyp	Ref	Expression
1		difundir 3390	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3493	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3288	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3458	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2202	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1353   \ cdif 3128   u. cun 3129   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425

This theorem is referenced by:  uneqdifeqim  3510  difprsn1  3733  orddif  4548  fisseneq  6934  dfn2  9192