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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 4485, 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 4246 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | ssinss1 4254 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐴) ⊆ 𝐶 |
4 | inssdif0 4380 | . 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 1537 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: resf1o 32748 gsummptres 33038 indsumin 34003 measunl 34197 carsgclctun 34303 probdif 34402 hgt750lemd 34642 redvmptabs 42369 |
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