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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 4266 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵) | |
2 | unidm 4130 | . . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵 | |
3 | 2 | difeq2i 4098 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtr3i 2848 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3935 ∪ cun 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 |
This theorem is referenced by: axcclem 9881 lpdifsn 21753 compne 40780 |
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