Theorem unidm 3293

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 758	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3292	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1364   e. wcel 2160   u. cun 3142

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148

This theorem is referenced by:  unundi  3311  unundir  3312  uneqin  3401  difabs  3414  ifidss  3564  dfsn2  3621  diftpsn3  3748  unisn  3840  dfdm2  5181  fun2  5408  resasplitss  5414  xpider  6633  pm54.43  7220