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Mirrors > Home > MPE Home > Th. List > Mathboxes > inindif | Structured version Visualization version GIF version |
Description: See inundif 4398. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
Ref | Expression |
---|---|
inindif | ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4149 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | 1 | orci 865 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) |
3 | inss 4158 | . . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
5 | inssdif0 4289 | . 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅) | |
6 | 4, 5 | mpbi 233 | 1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1543 ∖ cdif 3868 ∩ cin 3870 ⊆ wss 3871 ∅c0 4242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3415 df-dif 3874 df-in 3878 df-ss 3888 df-nul 4243 |
This theorem is referenced by: resf1o 30790 gsummptres 31036 indsumin 31707 measunl 31901 carsgclctun 32005 probdif 32104 hgt750lemd 32345 |
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