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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 4078 | . . 3 ⊢ (𝐵 ∪ (𝐶 ∪ 𝐷)) = (𝐶 ∪ (𝐵 ∪ 𝐷)) | |
2 | 1 | uneq2i 4071 | . 2 ⊢ (𝐴 ∪ (𝐵 ∪ (𝐶 ∪ 𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵 ∪ 𝐷))) |
3 | unass 4077 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶 ∪ 𝐷))) | |
4 | unass 4077 | . 2 ⊢ ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵 ∪ 𝐷))) | |
5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3861 |
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 3407 df-un 3868 |
This theorem is referenced by: unundi 4081 unundir 4082 xpun 5619 resasplit 6586 ex-pw 28509 iunrelexp0 40985 |
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