Theorem unidm 3276

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 757	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3275	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1353   e. wcel 2146   u. cun 3125

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131

This theorem is referenced by:  unundi  3294  unundir  3295  uneqin  3384  difabs  3397  ifidss  3547  dfsn2  3603  diftpsn3  3730  unisn  3821  dfdm2  5155  fun2  5381  resasplitss  5387  xpider  6596  pm54.43  7179