Theorem dfun3 3494

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 3491	. 2 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∖ B))
2		dfin2 3492	. . . 4 ⊢ ((V ∖ A) ∩ (V ∖ B)) = ((V ∖ A) ∖ (V ∖ (V ∖ B)))
3		ddif 3399	. . . . 5 ⊢ (V ∖ (V ∖ B)) = B
4	3	difeq2i 3383	. . . 4 ⊢ ((V ∖ A) ∖ (V ∖ (V ∖ B))) = ((V ∖ A) ∖ B)
5	2, 4	eqtr2i 2374	. . 3 ⊢ ((V ∖ A) ∖ B) = ((V ∖ A) ∩ (V ∖ B))
6	5	difeq2i 3383	. 2 ⊢ (V ∖ ((V ∖ A) ∖ B)) = (V ∖ ((V ∖ A) ∩ (V ∖ B)))
7	1, 6	eqtri 2373	1 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∩ (V ∖ B)))

This proof depends on syntax axioms:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is used by:  difundi  3508  undifv  3625