Theorem iinun 3548

Description: Complement of intersection is equal to union of complements. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
iinun	|- ∼ (A i^i B) = ( ∼ A u. ∼ B)

Proof of Theorem iinun

Step	Hyp	Ref	Expression
1		dfun4 3546	. 2 |- ( ∼ A u. ∼ B) = ∼ ( ∼ ∼ A i^i ∼ ∼ B)
2		dblcompl 3227	. . . 4 |- ∼ ∼ A = A
3		dblcompl 3227	. . . 4 |- ∼ ∼ B = B
4	2, 3	ineq12i 3455	. . 3 |- ( ∼ ∼ A i^i ∼ ∼ B) = (A i^i B)
5	4	compleqi 3244	. 2 |- ∼ ( ∼ ∼ A i^i ∼ ∼ B) = ∼ (A i^i B)
6	1, 5	eqtr2i 2374	1 |- ∼ (A i^i B) = ( ∼ A u. ∼ B)

Syntax hints:   = wceq 1642   ∼ ccompl 3205   u. cun 3207   i^i cin 3208

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214

This theorem is referenced by:  sbthlem1  6203