Theorem difundir 4237

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difundir	⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

Proof of Theorem difundir

Step	Hyp	Ref	Expression
1		indir 4232	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4225	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶)
3		invdif 4225	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
4		invdif 4225	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
5	3, 4	uneq12i 4113	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
6	1, 2, 5	3eqtr3i 2791	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906

This theorem is used by:  dfsymdif3  4252  difun2  4437  diftpsn3  4765  strleun  17274  setsfun0  17289  mreexmrid  17756  mreexexlem2d  17758  chnccats1  18738  mvdco  19598  dprd2da  20197  dmdprdsplit2lem  20200  ablfac1eulem  20227  lbsextlem4  21378  lindsenlbs  22096  opsrtoslem2  22304  nulmbl2  25796  uniioombllem3  25845  ltslpss  28205  leslss  28206  ex-dif  30935  indifundif  33031  imadifxp  33106  fzodif1  33295  cycpmrn  33615  ballotlemfp1  35036  ballotlemgun  35069  onint1  37135  lindsadd  38432  poimirlem2  38436  poimirlem6  38440  poimirlem7  38441  poimirlem8  38442  poimirlem22  38456  dmxrnuncnvepres  39205  dvmptfprodlem  46837  fourierdlem102  47101  fourierdlem114  47113  caragenuncllem  47405  carageniuncllem1  47414