Theorem dfin3 4230

Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfin3	⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Proof of Theorem dfin3

Step	Hyp	Ref	Expression
1		ddif 4095	. 2 ⊢ (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2		dfun2 4223	. . . 4 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3		ddif 4095	. . . . . 6 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴
4	3	difeq1i 4077	. . . . 5 ⊢ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
5	4	difeq2i 4078	. . . 4 ⊢ (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
6	2, 5	eqtri 2786	. . 3 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
7	6	difeq2i 4078	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8		dfin2 4224	. 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))
9	1, 7, 8	3eqtr4ri 2797	1 ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912

This theorem is referenced by:  difindi  4245