Theorem unidm 3316

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 759	. 2 |- ( ( x e. A \/ x e. A ) <-> x e. A )
2	1	uneqri 3315	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1373   e. wcel 2176   u. cun 3164

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170

This theorem is referenced by:  unundi  3334  unundir  3335  uneqin  3424  difabs  3437  ifidss  3586  dfsn2  3647  diftpsn3  3774  unisn  3866  dfdm2  5217  fun2  5449  resasplitss  5455  xpider  6693  pm54.43  7298  plyun0  15208