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Mirrors > Home > ILE Home > Th. List > inssdif0im | GIF version |
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.) |
Ref | Expression |
---|---|
inssdif0im | ⊢ ((A ∩ B) ⊆ 𝐶 → (A ∩ (B ∖ 𝐶)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3120 | . . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B)) | |
2 | 1 | imbi1i 227 | . . . . 5 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) ↔ ((x ∈ A ∧ x ∈ B) → x ∈ 𝐶)) |
3 | imanim 784 | . . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) → x ∈ 𝐶) → ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶)) | |
4 | 2, 3 | sylbi 114 | . . . 4 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) → ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶)) |
5 | eldif 2921 | . . . . . 6 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶)) | |
6 | 5 | anbi2i 430 | . . . . 5 ⊢ ((x ∈ A ∧ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ 𝐶))) |
7 | elin 3120 | . . . . 5 ⊢ (x ∈ (A ∩ (B ∖ 𝐶)) ↔ (x ∈ A ∧ x ∈ (B ∖ 𝐶))) | |
8 | anass 381 | . . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ 𝐶))) | |
9 | 6, 7, 8 | 3bitr4ri 202 | . . . 4 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶) ↔ x ∈ (A ∩ (B ∖ 𝐶))) |
10 | 4, 9 | sylnib 600 | . . 3 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) → ¬ x ∈ (A ∩ (B ∖ 𝐶))) |
11 | 10 | alimi 1341 | . 2 ⊢ (∀x(x ∈ (A ∩ B) → x ∈ 𝐶) → ∀x ¬ x ∈ (A ∩ (B ∖ 𝐶))) |
12 | dfss2 2928 | . 2 ⊢ ((A ∩ B) ⊆ 𝐶 ↔ ∀x(x ∈ (A ∩ B) → x ∈ 𝐶)) | |
13 | eq0 3233 | . 2 ⊢ ((A ∩ (B ∖ 𝐶)) = ∅ ↔ ∀x ¬ x ∈ (A ∩ (B ∖ 𝐶))) | |
14 | 11, 12, 13 | 3imtr4i 190 | 1 ⊢ ((A ∩ B) ⊆ 𝐶 → (A ∩ (B ∖ 𝐶)) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∖ cdif 2908 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-in 2918 df-ss 2925 df-nul 3219 |
This theorem is referenced by: disjdif 3290 |
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