| Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjdifb | Structured version Visualization version GIF version | ||
| Description: Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.) |
| Ref | Expression |
|---|---|
| disjdifb | ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | indif1 4282 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) | |
| 2 | disjdif 4472 | . . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
| 3 | 2 | difeq1i 4122 | . 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵) |
| 4 | 0dif 4405 | . 2 ⊢ (∅ ∖ 𝐵) = ∅ | |
| 5 | 1, 3, 4 | 3eqtri 2769 | 1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ∅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: iscnrm3r 48845 |
| Copyright terms: Public domain | W3C validator |