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Mirrors > Home > MPE Home > Th. List > difundi | Structured version Visualization version GIF version |
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
difundi | ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfun3 3898 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) | |
2 | 1 | difeq2i 3758 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) |
3 | inindi 3863 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) | |
4 | dfin2 3893 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) | |
5 | invdif 3901 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | invdif 3901 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
7 | 5, 6 | ineq12i 3845 | . . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
8 | 3, 4, 7 | 3eqtr3i 2681 | . 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
9 | 2, 8 | eqtri 2673 | 1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 Vcvv 3231 ∖ cdif 3604 ∪ cun 3605 ∩ cin 3606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 |
This theorem is referenced by: undm 3918 uncld 20893 inmbl 23356 difuncomp 29495 clsun 32448 poimirlem8 33547 ntrclskb 38684 ntrclsk3 38685 ntrclsk13 38686 |
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