Theorem unidm 3219

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 746	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	uneqri 3218	1 ⊢ (𝐴 ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1331   ∈ wcel 1480   ∪ cun 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075

This theorem is referenced by:  unundi  3237  unundir  3238  uneqin  3327  difabs  3340  dfsn2  3541  diftpsn3  3661  unisn  3752  dfdm2  5073  fun2  5296  resasplitss  5302  xpider  6500  pm54.43  7046