Theorem difundi 4182

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 4168	. . 3 ⊢ (𝐵 ∪ 𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2	1	difeq2i 4023	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3		inindi 4129	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4		dfin2 4163	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5		invdif 4171	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
6		invdif 4171	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
7	5, 6	ineq12i 4113	. . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
8	3, 4, 7	3eqtr3i 2829	. 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
9	2, 8	eqtri 2821	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1525  Vcvv 3440   ∖ cdif 3862   ∪ cun 3863   ∩ cin 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872

This theorem is referenced by:  undm  4188  uncld  21337  inmbl  23830  difuncomp  29990  clsun  33287  poimirlem8  34452  ntrclskb  39925  ntrclsk3  39926  ntrclsk13  39927