Theorem unidm 3144

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 710	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3143	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1290   e. wcel 1439   u. cun 2998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004

This theorem is referenced by:  unundi  3162  unundir  3163  uneqin  3251  difabs  3264  dfsn2  3464  diftpsn3  3584  unisn  3675  dfdm2  4978  fun2  5197  resasplitss  5203  xpiderm  6377  pm54.43  6879