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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 3875 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) | |
2 | invdif 3868 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶) | |
3 | invdif 3868 | . . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
4 | invdif 3868 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
5 | 3, 4 | uneq12i 3765 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
6 | 1, 2, 5 | 3eqtr3i 2652 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 |
This theorem is referenced by: dfsymdif3 3893 difun2 4048 diftpsn3 4332 diftpsn3OLD 4333 setsfun0 15894 strleun 15972 mreexmrid 16303 mreexexlem2d 16305 mvdco 17865 dprd2da 18441 dmdprdsplit2lem 18444 ablfac1eulem 18471 lbsextlem4 19161 opsrtoslem2 19485 nulmbl2 23304 uniioombllem3 23353 ex-dif 27280 indifundif 29356 imadifxp 29414 ballotlemfp1 30553 ballotlemgun 30586 onint1 32448 lindsenlbs 33404 poimirlem2 33411 poimirlem6 33415 poimirlem7 33416 poimirlem8 33417 poimirlem22 33431 dvmptfprodlem 40159 fourierdlem102 40425 fourierdlem114 40437 caragenuncllem 40726 carageniuncllem1 40735 |
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