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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 RP, 29-Feb-2020.) |
| Ref | Expression |
|---|---|
| rp-fakeuninass | ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rp-fakeinunass 43528 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴))) | |
| 2 | eqcom 2744 | . 2 ⊢ (((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵 ∪ 𝐴)) ↔ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴)) | |
| 3 | incom 4209 | . . . 4 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐵 ∪ 𝐴) ∩ 𝐶) | |
| 4 | uncom 4158 | . . . . 5 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
| 5 | 4 | ineq1i 4216 | . . . 4 ⊢ ((𝐵 ∪ 𝐴) ∩ 𝐶) = ((𝐴 ∪ 𝐵) ∩ 𝐶) |
| 6 | 3, 5 | eqtri 2765 | . . 3 ⊢ (𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ 𝐶) |
| 7 | uncom 4158 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶 ∩ 𝐵)) | |
| 8 | incom 4209 | . . . . 5 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
| 9 | 8 | uneq2i 4165 | . . . 4 ⊢ (𝐴 ∪ (𝐶 ∩ 𝐵)) = (𝐴 ∪ (𝐵 ∩ 𝐶)) |
| 10 | 7, 9 | eqtri 2765 | . . 3 ⊢ ((𝐶 ∩ 𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵 ∩ 𝐶)) |
| 11 | 6, 10 | eqeq12i 2755 | . 2 ⊢ ((𝐶 ∩ (𝐵 ∪ 𝐴)) = ((𝐶 ∩ 𝐵) ∪ 𝐴) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
| 12 | 1, 2, 11 | 3bitri 297 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 |
| This theorem is referenced by: (None) |
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