Theorem difun2 3530

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 3416	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3519	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3314	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3484	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2221	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1364   \ cdif 3154   u. cun 3155   (/)c0 3450

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by:  uneqdifeqim  3536  difprsn1  3761  orddif  4583  fisseneq  6995  dfn2  9262