Theorem unidm 3265

Description: Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
unidm	|- ( A u. A ) = A

Proof of Theorem unidm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oridm 747	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3264	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1343   e. wcel 2136   u. cun 3114

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-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-un 3120

This theorem is referenced by:  unundi  3283  unundir  3284  uneqin  3373  difabs  3386  ifidss  3535  dfsn2  3590  diftpsn3  3714  unisn  3805  dfdm2  5138  fun2  5361  resasplitss  5367  xpider  6572  pm54.43  7146