Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfin3 Structured version   Unicode version

Theorem dfin3 3718
 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

Proof of Theorem dfin3
StepHypRef Expression
1 ddif 3603 . 2
2 dfun2 3714 . . . 4
3 ddif 3603 . . . . . 6
43difeq1i 3585 . . . . 5
54difeq2i 3586 . . . 4
62, 5eqtri 2458 . . 3
76difeq2i 3586 . 2
8 dfin2 3715 . 2
91, 7, 83eqtr4ri 2469 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437  cvv 3087   cdif 3439   cun 3440   cin 3441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449 This theorem is referenced by:  difindi  3733
 Copyright terms: Public domain W3C validator