Theorem unidm 3224

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 3223	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1332   e. wcel 1481   u. cun 3074

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080

This theorem is referenced by:  unundi  3242  unundir  3243  uneqin  3332  difabs  3345  dfsn2  3546  diftpsn3  3669  unisn  3760  dfdm2  5081  fun2  5304  resasplitss  5310  xpider  6508  pm54.43  7063