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Theorem dfin5 3545
 Description: Definition of intersection in terms of union. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
dfin5 (AB) = ∼ ( ∼ A ∪ ∼ B)

Proof of Theorem dfin5
StepHypRef Expression
1 dblcompl 3227 . . . 4 ∼ ∼ A = A
2 dblcompl 3227 . . . 4 ∼ ∼ B = B
31, 2nineq12i 3239 . . 3 ( ∼ ∼ A ⩃ ∼ ∼ B) = (AB)
43compleqi 3244 . 2 ∼ ( ∼ ∼ A ⩃ ∼ ∼ B) = ∼ (AB)
5 df-un 3214 . . 3 ( ∼ A ∪ ∼ B) = ( ∼ ∼ A ⩃ ∼ ∼ B)
65compleqi 3244 . 2 ∼ ( ∼ A ∪ ∼ B) = ∼ ( ∼ ∼ A ⩃ ∼ ∼ B)
7 df-in 3213 . 2 (AB) = ∼ (AB)
84, 6, 73eqtr4ri 2384 1 (AB) = ∼ ( ∼ A ∪ ∼ B)
 Colors of variables: wff setvar class 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
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