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Mirrors > Home > MPE Home > Th. List > unundi | Structured version Visualization version GIF version |
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
unundi | ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unidm 3982 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
2 | 1 | uneq1i 3989 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
3 | un4 3999 | . 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) | |
4 | 2, 3 | eqtr3i 2850 | 1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∪ cun 3795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-v 3415 df-un 3802 |
This theorem is referenced by: dfif5 4321 |
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