Theorem unidm 3362

Description: Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
unidm	⊢ (𝐴 ∪ 𝐴) = 𝐴

Proof of Theorem unidm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oridm 765	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	uneqri 3361	1 ⊢ (𝐴 ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2203   ∪ cun 3209

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215

This theorem is referenced by:  unundi  3380  unundir  3381  uneqin  3472  difabs  3485  ifidss  3638  dfsn2  3703  diftpsn3  3835  unisn  3930  dfdm2  5297  fun2  5537  resasplitss  5544  xpider  6840  pm54.43  7487  plyun0  15601