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Mirrors > Home > MPE Home > Th. List > un12 | Structured version Visualization version 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 4128 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
2 | 1 | uneq1i 4134 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐵 ∪ 𝐴) ∪ 𝐶) |
3 | unass 4141 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
4 | unass 4141 | . 2 ⊢ ((𝐵 ∪ 𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2852 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 |
This theorem is referenced by: un23 4143 un4 4144 fresaun 6543 reconnlem1 23428 poimirlem6 34892 poimirlem7 34893 asindmre 34971 frege133d 40103 |
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