![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4283 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) | |
2 | 1 | difeq2i 4133 | . 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) |
3 | indi 4290 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) | |
4 | dfin2 4277 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) | |
5 | invdif 4285 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | invdif 4285 | . . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
7 | 5, 6 | uneq12i 4176 | . . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
8 | 3, 4, 7 | 3eqtr3i 2771 | . 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
9 | 2, 8 | eqtri 2763 | 1 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 |
This theorem is referenced by: difdif2 4302 indm 4304 fndifnfp 7196 dprddisj2 20074 fctop 23027 cctop 23029 mretopd 23116 restcld 23196 cfinfil 23917 csdfil 23918 indifundif 32552 difres 32620 unelcarsg 34294 clsk3nimkb 44030 ntrclskb 44059 ntrclsk3 44060 ntrclsk13 44061 salincl 46280 iscnrm3rlem1 48737 |
Copyright terms: Public domain | W3C validator |