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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 4166 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) | |
2 | 1 | difeq2i 4020 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) |
3 | inindi 4127 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) | |
4 | dfin2 4161 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) | |
5 | invdif 4169 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | invdif 4169 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
7 | 5, 6 | ineq12i 4111 | . . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
8 | 3, 4, 7 | 3eqtr3i 2767 | . 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
9 | 2, 8 | eqtri 2759 | 1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 ∩ cin 3852 |
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: undm 4188 uncld 21892 inmbl 24393 difuncomp 30566 clsun 34203 poimirlem8 35471 ntrclskb 41297 ntrclsk3 41298 ntrclsk13 41299 |
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