Theorem difun2 3488

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 3375	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3477	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3273	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3442	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2190	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1343   \ cdif 3113   u. cun 3114   (/)c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by:  uneqdifeqim  3494  difprsn1  3712  orddif  4524  fisseneq  6897  dfn2  9127