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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inindif | Structured version Visualization version GIF version |
Description: See inundif 4385. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
Ref | Expression |
---|---|
inindif | ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4156 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | 1 | orci 862 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) |
3 | inss 4165 | . . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
5 | inssdif0 4283 | . 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 844 = wceq 1538 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 |
This theorem is referenced by: resf1o 30492 gsummptres 30737 indsumin 31391 measunl 31585 carsgclctun 31689 probdif 31788 hgt750lemd 32029 |
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