Theorem unidm 3347

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 762	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
2	1	uneqri 3346	1 ⊢ (𝐴 ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200   ∪ cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  unundi  3365  unundir  3366  uneqin  3455  difabs  3468  ifidss  3618  dfsn2  3680  diftpsn3  3808  unisn  3903  dfdm2  5262  fun2  5497  resasplitss  5504  xpider  6751  pm54.43  7359  plyun0  15404