Theorem dfin3 3495

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 3399	. 2 ⊢ (V ∖ (V ∖ (A ∖ (V ∖ B)))) = (A ∖ (V ∖ B))
2		dfun2 3491	. . . 4 ⊢ ((V ∖ A) ∪ (V ∖ B)) = (V ∖ ((V ∖ (V ∖ A)) ∖ (V ∖ B)))
3		ddif 3399	. . . . . 6 ⊢ (V ∖ (V ∖ A)) = A
4	3	difeq1i 3382	. . . . 5 ⊢ ((V ∖ (V ∖ A)) ∖ (V ∖ B)) = (A ∖ (V ∖ B))
5	4	difeq2i 3383	. . . 4 ⊢ (V ∖ ((V ∖ (V ∖ A)) ∖ (V ∖ B))) = (V ∖ (A ∖ (V ∖ B)))
6	2, 5	eqtri 2373	. . 3 ⊢ ((V ∖ A) ∪ (V ∖ B)) = (V ∖ (A ∖ (V ∖ B)))
7	6	difeq2i 3383	. 2 ⊢ (V ∖ ((V ∖ A) ∪ (V ∖ B))) = (V ∖ (V ∖ (A ∖ (V ∖ B))))
8		dfin2 3492	. 2 ⊢ (A ∩ B) = (A ∖ (V ∖ B))
9	1, 7, 8	3eqtr4ri 2384	1 ⊢ (A ∩ B) = (V ∖ ((V ∖ A) ∪ (V ∖ B)))

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This theorem was proved from 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 theorem 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 referenced by:  difindi  3510