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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 4276 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) | |
2 | invdif 4269 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶) | |
3 | invdif 4269 | . . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶) | |
4 | invdif 4269 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
5 | 3, 4 | uneq12i 4162 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
6 | 1, 2, 5 | 3eqtr3i 2769 | 1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 |
This theorem is referenced by: dfsymdif3 4297 difun2 4481 diftpsn3 4806 strleun 17090 setsfun0 17105 mreexmrid 17587 mreexexlem2d 17589 mvdco 19313 dprd2da 19912 dmdprdsplit2lem 19915 ablfac1eulem 19942 lbsextlem4 20774 opsrtoslem2 21617 nulmbl2 25053 uniioombllem3 25102 sltlpss 27401 ex-dif 29676 indifundif 31762 imadifxp 31832 fzodif1 32004 cycpmrn 32302 ballotlemfp1 33490 ballotlemgun 33523 onint1 35334 lindsadd 36481 lindsenlbs 36483 poimirlem2 36490 poimirlem6 36494 poimirlem7 36495 poimirlem8 36496 poimirlem22 36510 dvmptfprodlem 44660 fourierdlem102 44924 fourierdlem114 44936 caragenuncllem 45228 carageniuncllem1 45237 |
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