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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 4190 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵) | |
2 | unidm 4052 | . . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵 | |
3 | 2 | difeq2i 4020 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtr3i 2761 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∖ cdif 3850 ∪ cun 3851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 |
This theorem is referenced by: axcclem 10036 lpdifsn 21994 compne 41673 |
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