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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 4178 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) | |
2 | disjdif 4371 | . . 3 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
3 | 2 | difeq1i 4026 | . 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵) |
4 | 0dif 4300 | . 2 ⊢ (∅ ∖ 𝐵) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2785 | 1 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3857 ∩ cin 3859 ∅c0 4227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-rab 3079 df-v 3411 df-dif 3863 df-in 3867 df-ss 3877 df-nul 4228 |
This theorem is referenced by: iscnrm3r 45681 |
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