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Mirrors > Home > MPE Home > Th. List > difabs | Structured version Visualization version GIF version |
Description: Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.) |
Ref | Expression |
---|---|
difabs | ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difun1 4288 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵) | |
2 | unidm 4149 | . . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵 | |
3 | 2 | difeq2i 4115 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtr3i 2755 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3941 ∪ cun 3942 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 |
This theorem is referenced by: axcclem 10487 lpdifsn 23108 compne 44025 |
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