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Mirrors > Home > MPE Home > Th. List > un4 | Structured version Visualization version GIF version |
Description: A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
un4 | ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un12 4159 | . . 3 ⊢ (𝐵 ∪ (𝐶 ∪ 𝐷)) = (𝐶 ∪ (𝐵 ∪ 𝐷)) | |
2 | 1 | uneq2i 4152 | . 2 ⊢ (𝐴 ∪ (𝐵 ∪ (𝐶 ∪ 𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵 ∪ 𝐷))) |
3 | unass 4158 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶 ∪ 𝐷))) | |
4 | unass 4158 | . 2 ⊢ ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵 ∪ 𝐷))) | |
5 | 2, 3, 4 | 3eqtr4i 2762 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-un 3945 |
This theorem is referenced by: unundi 4162 unundir 4163 xpun 5739 resasplit 6751 addsdi 27971 mulsass 27982 ex-pw 30151 iunrelexp0 42942 |
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