Theorem difundi 3879

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 3865	. . 3 ⊢ (𝐵 ∪ 𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2	1	difeq2i 3725	. 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3		inindi 3830	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4		dfin2 3860	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5		invdif 3868	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
6		invdif 3868	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
7	5, 6	ineq12i 3812	. . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
8	3, 4, 7	3eqtr3i 2652	. 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
9	2, 8	eqtri 2644	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

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:  undm  3885  uncld  20845  inmbl  23310  difuncomp  29369  clsun  32323  poimirlem8  33417  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369