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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 | pm4.61 405 | . . 3 ⊢ (¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | anclb 546 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
3 | elin 4168 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | 3 | imbi2i 337 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
5 | iman 402 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵))) | |
6 | 2, 4, 5 | 3bitr2i 300 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵))) |
7 | 6 | con2bii 359 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
8 | eldif 3945 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
9 | 1, 7, 8 | 3bitr4i 304 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) |
10 | 9 | difeqri 4100 | 1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3932 ∩ cin 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3938 df-in 3942 |
This theorem is referenced by: dfin4 4243 indif 4245 dfsymdif3 4268 notrab 4279 disjdif2 4426 dfsdom2 8629 hashdif 13764 isercolllem3 15013 iuncld 21583 llycmpkgen2 22088 1stckgen 22092 txkgen 22190 cmmbl 24064 disjdifprg2 30255 ldgenpisyslem1 31322 onint1 33695 nonrel 39824 nzprmdif 40531 |
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