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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 4196 | . 2 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐵 ∩ (𝐴 ∩ 𝐴)) | |
2 | inidm 4192 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
3 | 2 | ineq2i 4183 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐴)) = (𝐵 ∩ 𝐴) |
4 | incom 4175 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2845 | 1 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∩ cin 3932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 |
This theorem is referenced by: measinb2 31381 |
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