Theorem difun2 3368

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 3255	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3357	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3154	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3322	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2113	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1290   \ cdif 2999   u. cun 3000   (/)c0 3289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290

This theorem is referenced by:  uneqdifeqim  3374  difprsn1  3584  orddif  4378  fisseneq  6698  dfn2  8749