Theorem dfun3 4222

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 4216	. 2 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
2		dfin2 4217	. . . 4 ⊢ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵)))
3		ddif 4088	. . . . 5 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
4	3	difeq2i 4071	. . . 4 ⊢ ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵)
5	2, 4	eqtr2i 2784	. . 3 ⊢ ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
6	5	difeq2i 4071	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
7	1, 6	eqtri 2783	1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906

This theorem is used by:  difundi  4236  unvdif  4429