| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > inin | Structured version Visualization version GIF version | ||
| Description: Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.) |
| Ref | Expression |
|---|---|
| inin | ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | in13 4231 | . 2 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐵 ∩ (𝐴 ∩ 𝐴)) | |
| 2 | inidm 4227 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 3 | 2 | ineq2i 4217 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐴)) = (𝐵 ∩ 𝐴) |
| 4 | incom 4209 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
| 5 | 1, 3, 4 | 3eqtri 2769 | 1 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∩ cin 3950 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 |
| This theorem is referenced by: measinb2 34224 |
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