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Mirrors > Home > ILE Home > Th. List > unidm | GIF version |
Description: Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
unidm | ⊢ (𝐴 ∪ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oridm 707 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
2 | 1 | uneqri 3115 | 1 ⊢ (𝐴 ∪ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∈ wcel 1434 ∪ cun 2972 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-un 2978 |
This theorem is referenced by: unundi 3134 unundir 3135 uneqin 3222 difabs 3235 dfsn2 3420 diftpsn3 3535 unisn 3625 dfdm2 4882 fun2 5095 resasplitss 5100 xpiderm 6243 pm54.43 6518 |
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