Theorem unidm 3366

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 765	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3365	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1398   e. wcel 2205   u. cun 3212

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

This theorem is referenced by:  unundi  3384  unundir  3385  uneqin  3476  difabs  3489  ifidss  3642  dfsn2  3708  diftpsn3  3840  unisn  3935  dfdm2  5302  fun2  5542  resasplitss  5549  xpider  6853  pm54.43  7500  plyun0  15727