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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 4080 | . . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | |
2 | 1 | uneq1i 4086 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐵 ∪ 𝐴) ∪ 𝐶) |
3 | unass 4093 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
4 | unass 4093 | . 2 ⊢ ((𝐵 ∪ 𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2829 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 |
This theorem is referenced by: un23 4095 un4 4096 fresaun 6523 reconnlem1 23431 poimirlem6 35063 poimirlem7 35064 asindmre 35140 frege133d 40466 |
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