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Mirrors > Home > MPE Home > Th. List > Mathboxes > inabs3 | Structured version Visualization version GIF version |
Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
inabs3 | ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4195 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
2 | sseqin2 4191 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶) | |
3 | 2 | biimpi 218 | . . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) = 𝐶) |
4 | 3 | ineq2d 4188 | . 2 ⊢ (𝐶 ⊆ 𝐵 → (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ 𝐶)) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3934 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 |
This theorem is referenced by: carageniuncllem1 42802 |
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