Theorem difun2 3540

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 3426	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3529	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3324	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3494	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2230	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1373   \ cdif 3163   u. cun 3164   (/)c0 3460

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  uneqdifeqim  3546  difprsn1  3772  orddif  4596  fisseneq  7033  dfn2  9310