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Mirrors > Home > MPE Home > Th. List > unundir | Structured version Visualization version GIF version |
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
unundir | ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unidm 4052 | . . 3 ⊢ (𝐶 ∪ 𝐶) = 𝐶 | |
2 | 1 | uneq2i 4060 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ 𝐶) |
3 | un4 4069 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) | |
4 | 2, 3 | eqtr3i 2761 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3851 |
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 2018 ax-8 2114 ax-9 2122 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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 |
This theorem is referenced by: iocunico 40686 |
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