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Mirrors > Home > MPE Home > Th. List > difindi | Structured version Visualization version GIF version |
Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
difindi | ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin3 4265 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) | |
2 | 1 | difeq2i 4115 | . 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) |
3 | indi 4272 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) | |
4 | dfin2 4259 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) | |
5 | invdif 4267 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | invdif 4267 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
7 | 5, 6 | uneq12i 4158 | . . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
8 | 3, 4, 7 | 3eqtr3i 2761 | . 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
9 | 2, 8 | eqtri 2753 | 1 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3461 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 |
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: difdif2 4285 indm 4287 fndifnfp 7185 dprddisj2 20008 fctop 22951 cctop 22953 mretopd 23040 restcld 23120 cfinfil 23841 csdfil 23842 indifundif 32400 difres 32469 unelcarsg 34063 clsk3nimkb 43612 ntrclskb 43641 ntrclsk3 43642 ntrclsk13 43643 salincl 45850 iscnrm3rlem1 48145 |
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