Theorem unidm 3307

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 3306	1 |- ( A u. A ) = A

Syntax hints:   = wceq 1364   e. wcel 2167   u. cun 3155

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161

This theorem is referenced by:  unundi  3325  unundir  3326  uneqin  3415  difabs  3428  ifidss  3577  dfsn2  3637  diftpsn3  3764  unisn  3856  dfdm2  5205  fun2  5434  resasplitss  5440  xpider  6674  pm54.43  7269  plyun0  15056