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Mirrors > Home > MPE Home > Th. List > un23 | Structured version Visualization version GIF version |
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
un23 | ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unass 4080 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
2 | un12 4081 | . 2 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
3 | uncom 4067 | . 2 ⊢ (𝐵 ∪ (𝐴 ∪ 𝐶)) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | |
4 | 1, 2, 3 | 3eqtri 2769 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 |
This theorem is referenced by: ssunpr 4745 setscom 16733 cycpmco2rn 31111 poimirlem6 35520 poimirlem7 35521 poimirlem16 35530 poimirlem19 35533 iocunico 40745 dfrcl2 40959 |
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