Theorem dfin5 3545

Description: Definition of intersection in terms of union. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
dfin5	⊢ (A ∩ B) = ∼ ( ∼ A ∪ ∼ B)

Proof of Theorem dfin5

Step	Hyp	Ref	Expression
1		dblcompl 3227	. . . 4 ⊢ ∼ ∼ A = A
2		dblcompl 3227	. . . 4 ⊢ ∼ ∼ B = B
3	1, 2	nineq12i 3239	. . 3 ⊢ ( ∼ ∼ A ⩃ ∼ ∼ B) = (A ⩃ B)
4	3	compleqi 3244	. 2 ⊢ ∼ ( ∼ ∼ A ⩃ ∼ ∼ B) = ∼ (A ⩃ B)
5		df-un 3214	. . 3 ⊢ ( ∼ A ∪ ∼ B) = ( ∼ ∼ A ⩃ ∼ ∼ B)
6	5	compleqi 3244	. 2 ⊢ ∼ ( ∼ A ∪ ∼ B) = ∼ ( ∼ ∼ A ⩃ ∼ ∼ B)
7		df-in 3213	. 2 ⊢ (A ∩ B) = ∼ (A ⩃ B)
8	4, 6, 7	3eqtr4ri 2384	1 ⊢ (A ∩ B) = ∼ ( ∼ A ∪ ∼ B)

Syntax hints:   = wceq 1642   ⩃ cnin 3204   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  dfun4  3546  iunin  3547