Theorem difundi 4255

Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difundi	⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Proof of Theorem difundi

Step	Hyp	Ref	Expression
1		dfun3 4241	. . 3 ⊢ (𝐵 ∪ 𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2	1	difeq2i 4095	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3		inindi 4202	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4		dfin2 4236	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5		invdif 4244	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
6		invdif 4244	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
7	5, 6	ineq12i 4186	. . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
8	3, 4, 7	3eqtr3i 2852	. 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
9	2, 8	eqtri 2844	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1533  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934

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 2157  ax-12 2173  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 3496  df-dif 3938  df-un 3940  df-in 3942

This theorem is referenced by:  undm  4261  uncld  21643  inmbl  24137  difuncomp  30299  clsun  33671  poimirlem8  34894  ntrclskb  40412  ntrclsk3  40413  ntrclsk13  40414