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Mirrors > Home > MPE Home > Th. List > difin | Structured version Visualization version GIF version |
Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difin | ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.61 404 | . . 3 ⊢ (¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | anclb 545 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
3 | elin 3978 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | 3 | imbi2i 336 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
5 | iman 401 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵))) | |
6 | 2, 4, 5 | 3bitr2i 299 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵))) |
7 | 6 | con2bii 357 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
8 | eldif 3972 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
9 | 1, 7, 8 | 3bitr4i 303 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) |
10 | 9 | difeqri 4137 | 1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ∩ cin 3961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 df-in 3969 |
This theorem is referenced by: dfin4 4283 indif 4285 dfsymdif3 4311 notrab 4327 disjdif2 4485 dfsdom2 9134 hashdif 14448 isercolllem3 15699 iuncld 23068 llycmpkgen2 23573 1stckgen 23577 txkgen 23675 cmmbl 25582 indifbi 32547 disjdifprg2 32595 ldgenpisyslem1 34143 onint1 36431 nonrel 43573 nzprmdif 44314 |
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