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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 4264 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) | |
2 | disjdif 4464 | . . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
3 | 2 | difeq1i 4111 | . 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵) |
4 | 0dif 4394 | . 2 ⊢ (∅ ∖ 𝐵) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2756 | 1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3938 ∩ cin 3940 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 |
This theorem is referenced by: iscnrm3r 47793 |
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