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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 4202 | . 2 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐵 ∩ (𝐴 ∩ 𝐴)) | |
2 | inidm 4198 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
3 | 2 | ineq2i 4189 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐴)) = (𝐵 ∩ 𝐴) |
4 | incom 4181 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
5 | 1, 3, 4 | 3eqtri 2851 | 1 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∩ cin 3938 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-in 3946 |
This theorem is referenced by: measinb2 31486 |
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