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| Mirrors > Home > MPE Home > Th. List > inindif | Structured version Visualization version GIF version | ||
| Description: The intersection and class difference of a class with another class are disjoint. With inundif 4479, this shows that such intersection and class difference partition the class 𝐴. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
| Ref | Expression |
|---|---|
| inindif | ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inss2 4238 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
| 2 | ssinss1 4246 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
| 4 | inssdif0 4374 | . 2 ⊢ (((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅) | |
| 5 | 3, 4 | mpbi 230 | 1 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 |
| This theorem is referenced by: resf1o 32741 indsumin 32847 gsummptres 33055 measunl 34217 carsgclctun 34323 probdif 34422 hgt750lemd 34663 redvmptabs 42390 |
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