Theorem unabs 3456

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

Assertion

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

Proof of Theorem unabs

Step	Hyp	Ref	Expression
1		inss1 3445	. 2 |- ( A i^i B ) C_ A
2		ssequn2 3396	. 2 |- ( ( A i^i B ) C_ A <-> ( A u. ( A i^i B ) ) = A )
3	1, 2	mpbi 145	1 |- ( A u. ( A i^i B ) ) = A

Syntax hints:   = wceq 1398   u. cun 3212   i^i cin 3213   C_ wss 3214

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227

This theorem is referenced by: (None)