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Mirrors > Home > MPE Home > Th. List > Mathboxes > inindif | Structured version Visualization version GIF version |
Description: See inundif 3998. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
Ref | Expression |
---|---|
inindif | ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3796 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | 1 | orci 404 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) |
3 | inss 3804 | . . 3 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
5 | inssdif0 3901 | . 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅) | |
6 | 4, 5 | mpbi 219 | 1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1475 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: resf1o 28893 gsummptres 29115 measunl 29606 carsgclctun 29710 probdif 29809 |
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