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Mirrors > Home > MPE Home > Th. List > difundir | Structured version Visualization version GIF version |
Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
difundir | ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indir 4252 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) | |
2 | invdif 4245 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶) | |
3 | invdif 4245 | . . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
4 | invdif 4245 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
5 | 3, 4 | uneq12i 4137 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
6 | 1, 2, 5 | 3eqtr3i 2852 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3495 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 |
This theorem is referenced by: dfsymdif3 4269 difun2 4429 diftpsn3 4729 setsfun0 16513 strleun 16585 mreexmrid 16908 mreexexlem2d 16910 mvdco 18567 dprd2da 19158 dmdprdsplit2lem 19161 ablfac1eulem 19188 lbsextlem4 19927 opsrtoslem2 20259 nulmbl2 24131 uniioombllem3 24180 ex-dif 28196 indifundif 30279 imadifxp 30345 fzodif1 30510 cycpmrn 30780 ballotlemfp1 31744 ballotlemgun 31777 onint1 33792 lindsadd 34879 lindsenlbs 34881 poimirlem2 34888 poimirlem6 34892 poimirlem7 34893 poimirlem8 34894 poimirlem22 34908 dvmptfprodlem 42221 fourierdlem102 42486 fourierdlem114 42498 caragenuncllem 42787 carageniuncllem1 42796 |
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