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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 3757 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
2 | 1 | uneq1i 3763 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐵 ∪ 𝐴) ∪ 𝐶) |
3 | unass 3770 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
4 | unass 3770 | . 2 ⊢ ((𝐵 ∪ 𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2652 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 |
This theorem is referenced by: un23 3772 un4 3773 fresaun 6075 reconnlem1 22629 poimirlem6 33415 poimirlem7 33416 asindmre 33495 frege133d 38057 |
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