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Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-fakeuninass | Structured version Visualization version GIF version |
Description: A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.) |
Ref | Expression |
---|---|
rp-fakeuninass | ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rp-fakeinunass 36880 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴))) | |
2 | eqcom 2617 | . 2 ⊢ (((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴)) ↔ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴)) | |
3 | incom 3767 | . . . 4 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐵 ∪ 𝐴) ∩ 𝐶) | |
4 | uncom 3719 | . . . . 5 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
5 | 4 | ineq1i 3772 | . . . 4 ⊢ ((𝐵 ∪ 𝐴) ∩ 𝐶) = ((𝐴 ∪ 𝐵) ∩ 𝐶) |
6 | 3, 5 | eqtri 2632 | . . 3 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ 𝐶) |
7 | uncom 3719 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶 ∩ 𝐵)) | |
8 | incom 3767 | . . . . 5 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
9 | 8 | uneq2i 3726 | . . . 4 ⊢ (𝐴 ∪ (𝐶 ∩ 𝐵)) = (𝐴 ∪ (𝐵 ∩ 𝐶)) |
10 | 7, 9 | eqtri 2632 | . . 3 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵 ∩ 𝐶)) |
11 | 6, 10 | eqeq12i 2624 | . 2 ⊢ ((𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
12 | 1, 2, 11 | 3bitri 285 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 |
This theorem is referenced by: (None) |
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