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Mirrors > Home > MPE Home > Th. List > unundi | Structured version Visualization version GIF version |
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
unundi | ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unidm 3718 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
2 | 1 | uneq1i 3725 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
3 | un4 3735 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) | |
4 | 2, 3 | eqtr3i 2634 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 |
This theorem is referenced by: dfif5 4052 |
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