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Mirrors > Home > MPE Home > Th. List > undir | Structured version Visualization version GIF version |
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
undir | ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undi 4291 | . 2 ⊢ (𝐶 ∪ (𝐴 ∩ 𝐵)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵)) | |
2 | uncom 4168 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴 ∩ 𝐵)) | |
3 | uncom 4168 | . . 3 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴) | |
4 | uncom 4168 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
5 | 3, 4 | ineq12i 4226 | . 2 ⊢ ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2773 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 |
This theorem is referenced by: undif1 4482 dfif4 4546 dfif5 4547 bwth 23434 |
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