Theorem inabs 3436

Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Assertion

Ref	Expression
inabs	|- ( A i^i ( A u. B ) ) = A

Proof of Theorem inabs

Step	Hyp	Ref	Expression
1		ssun1 3367	. 2 |- A C_ ( A u. B )
2		df-ss 3210	. 2 |- ( A C_ ( A u. B ) <-> ( A i^i ( A u. B ) ) = A )
3	1, 2	mpbi 145	1 |- ( A i^i ( A u. B ) ) = A

Syntax hints:   = wceq 1395   u. cun 3195   i^i cin 3196   C_ wss 3197

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by: (None)