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Mirrors > Home > ILE Home > Th. List > un12 | GIF version |
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
un12 | ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3167 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
2 | 1 | uneq1i 3173 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐵 ∪ 𝐴) ∪ 𝐶) |
3 | unass 3180 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
4 | unass 3180 | . 2 ⊢ ((𝐵 ∪ 𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2128 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∪ cun 3019 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 |
This theorem is referenced by: un23 3182 un4 3183 |
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