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Theorem dfun3 3493
 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 (AB) = (V ((V A) ∩ (V B)))

Proof of Theorem dfun3
StepHypRef Expression
1 dfun2 3490 . 2 (AB) = (V ((V A) B))
2 dfin2 3491 . . . 4 ((V A) ∩ (V B)) = ((V A) (V (V B)))
3 ddif 3398 . . . . 5 (V (V B)) = B
43difeq2i 3382 . . . 4 ((V A) (V (V B))) = ((V A) B)
52, 4eqtr2i 2374 . . 3 ((V A) B) = ((V A) ∩ (V B))
65difeq2i 3382 . 2 (V ((V A) B)) = (V ((V A) ∩ (V B)))
71, 6eqtri 2373 1 (AB) = (V ((V A) ∩ (V B)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ 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  df-dif 3215 This theorem is referenced by:  difundi  3507  undifv  3624
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