Theorem dfun3 4228

Description: Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfun3	⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Proof of Theorem dfun3

Step	Hyp	Ref	Expression
1		dfun2 4222	. 2 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
2		dfin2 4223	. . . 4 ⊢ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵)))
3		ddif 4094	. . . . 5 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
4	3	difeq2i 4077	. . . 4 ⊢ ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵)
5	2, 4	eqtr2i 2786	. . 3 ⊢ ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
6	5	difeq2i 4077	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
7	1, 6	eqtri 2785	1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Syntax hints:   = wceq 1560  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911

This theorem is referenced by:  difundi  4242  unvdif  4429